Calculus IV - HW 1. Section 20. Due 6/16
|
|
- Lester Barnett
- 6 years ago
- Views:
Transcription
1 Calculus IV - HW Section 0 Due 6/6 Section.. Given both of the equations y = 4 y and y = 3y 3, draw a direction field for the differential equation. Based on the direction field, determine the behavior of y as t. Does this behavior depend on the initial value of at t=0? If so, how? Solution: In this case y approaches as t approaches infinity. This is independent of the initial value of y(0). Here there are three cases. If y(0) = Then y is constant and y remains constant as t approaches infinity. If y(0) < then y as t. If y(0) > then y as t
2 . (a) Give two different differential equations of the form dy dt solutions approach y = 5. = ay + b such that all Solution: The general solution to dy = ay + b is y = dt ceat b. It follows that a if a < 0 and b = 5 then the solution will behave as specified in the question. a Two examples are dy dy = y + 5 and = y + 0. dt dt (b) Give two differential equation of the form dy dt diverge from y =. = ay + b such that all solutions Solution: By similar logic to the above if a > 0 and b = the solution will a satisfy the above conditions. Two examples are dy dy = y and = 4y. dt dt 3. Problem 5-0 from section. of Boyce and Diprima. Solution: Found in the back of B&D. 4. Optional Hard Problem (Problem from B&D) A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential equation for the volume of the raindrop as a function of time. Solution: The equation for the volume of a sphere is V = 4 3 πr3 and the equation for the surface area is S = 4πr. Solving the volume equation for r we have ) r = ( 3V 3. Plugging into the equation for Surface area we have S = π V 3. We 4π are given that dv = ks where the negative sign is used (really as a notational dt device) because the volume is decreasing and has a negative derivative with respect to time. Putting it all together yields dv = kv 3. Here k = π k. It is common dt to continually absorb all constants into one undetermined constant term, and often we don t even bother to rename the constants as we do this. So the final solution is dv = kv 3. dt Page
3 Section. 5. (Problem 3 from B&D) Consider the differential equation dy dt and b are positive numbers. = ay + b where both a (a) Solve the differential equation. dy Solution: Some algebraic manipulation gives = a. We can then integrate both sides to get ln( y b ) = at + c and y b = a a ±e at+c. y b dt a The final solution is then y = ce at + b where c can be any real number. a Notice that the relative value of the constant c changed throughout the solution process, and the reason we don t take care to keep track of the exact value is the following. Given an initial condition the final answer will come out to be the same regardless of how c is represented. It is worth mentioning, though, why c can be any real number. Notice first that e x ranges over all positive real numbers so ±e x ranges over all nonzero real numbers. Where do we get 0 as a possibility in our final solution? When c = 0 we have the equilibrium solution, which (think about why this happened) we lost during the solving process. (b) Sketch the solution for several different initial conditions. Solution: Here are two variations plotted. Page 3
4 (c) Describe how the solutions change under each of the following conditions: i. a increases Solution: If a increases the value that each solution converges to decreases and the rate at which the solutions converge increases. ii. b increases Solution: If b increases the value that each solution converges to increases. iii. Both a and b increase, but the ration b a remains the same. Solution: If both a and b increase, but the ration b remains the same a then the rate at which the solutions converge increases. 6. (Problem 7 from B&D) Suppose a field mouse population satisfies the differential equation dp = 0.5p 450 dt (a) Find the time at which the population becomes extinct if p(0) = 850. Solution: As mentioned above, the general solution to dy dt = ay + b is y = ce at b a. Therefore, the solution in this case is p(t) = ce.5t Plugging the initial condition into the equation yields 850 = c and c = 50. So we want to solve p(s) = 0 for s. That is 0 = 50e.5t which happens when t = ln(8) months. (b) Find the time of extinction if p(0) = p 0, where 0 < p 0 < 900. Solution: Plugging the initial condition into p(t) = ce.5t yields p 0 = c and c = p In order to find the time of extinction, we ( need ) to solve P (T ) = 0 for T. That is 0 = (p 0 900)e.5T +900 and T = ln. 900 p Note that the initial restriction on p 0 is necessary here so that the term inside the logarithm is positive. (c) Find the initial population p 0 if the population is to become extinct in year. Page 4
5 Solution: For this we need to solve P () = 0 for p 0. Here we are assuming that t is in months, but if you assumed that t was in years or days or scores that s fine because we didn t specify. We then have 0 = (p 0 900)e and p 0 = e (Problem 8 from B&D) Consider a population p of field mice that rows at a rate proportional to the current population, so that dp dt = rp (a) Find the rate constant r if the population doubles in 30 days. Solution: Solving the differential equation gives p(t) = ce rt. Assuming t is in days we are trying to find r so that p(30) = p(0). That is ce 30r = ce 0r = c and r = ln() (b) Find r if the population doubles in N days. Solution: Following the same steps as above you should find the solution to be ln() N. Section.3 8. For each of the following, determine the order of the differential equation and whether the equation is linear or nonlinear. (a) t 3 d y dt + (t + ) dy dt + 5y = sin(t) Solution: Linear, order (b) y d3 y dt 3 + dy dt = t Solution: Nonlinear, order 3 (c) dy dt + sin(y) = 0 Solution: Nonlinear, order Page 5
6 (d) d y + sin(t + )y = 5t dt Solution: Linear, order 9. (Problem 9-0 from B&D) For each of the following, verify that the given function is a solution of the differential equation. (a) ty y = t ; y = 3t + t Solution: Noting that y = 3+t and plugging in, we can verify that ty y = t(3 + t) (3t + t ) = t (b) y + 4y + 3y = t ; y (t) = t 3, y (t) = e t + t 3 Solution: Notice the following. y (t) = t 3 y (t) = 3 y (t) = y (t) = y (t) = 0 From the above it is clear that y is a solution. Now let y = e t and notice that y + 4y + 3y = 0. By linearity of differentiation we have that e t + t 3 is also a solution to the above differential equation. 0. (Problem 7 from B&D) Determine the values of r for which y + y 6y has solutions of the form y = e rt. Solution: Plugging e rt into the equation, we need r so that r e rt + re rt 6e rt = 0. Because e rt is never zero we can divide through by it and are left with r +r 6 = 0. It follows that r = 3 and r = are the only solutions that work.. For each part below, give an example of an ODE satisfying the stated conditions: (a) nd order linear equation (b) 5th order linear equation involving a trig function (c) 3rd order nonlinear equation involving a trig function. (d) 4th order nonlinear equation not involving a trig function Page 6
7 Solution: Many correct solutions exist but here are a few: (a) y + y + 5y = 6 (b) d5 y dt 5 + sin(t) dy dt + y = t (c) y + ty = sin(y) (d) d4 y dt 4 + y d y dt + y = 0 Section. Each answer must be supported by a sufficient amount of work. Because many of the answers to these questions are in the back of your textbook, answers without work will not receive any credit.. Consider the differential equation µ (t) = µ(t)p(t) where p(t) is any integrable function and µ(t) > 0 for all t R. (a) Solve the differential equation by first dividing both sides by µ(t) (it is given that it is never 0), then integrating both sides, and finally exponentiating both sides. Solution: Following the steps outlined above we have µ (t) µ(t) = p(t). Integrating yields ln(µ(t)) = p(t) dt. We do not need the absolute value within the logarithm because µ(t) is positive everywhere. Exponentiating gives and completes the solution process. µ(t) = e p(t) dt Page 7
8 (b) Determine µ(t)y (t) + µ(t)p(t)y(t) dt. (Hint what is µ(t)p(t) equal to?) Solution: Noticing that µ(t)p(t) = µ (t) we have µ(t)y (t) + µ (t)y(t) dt. Considering the product rule for differentiating, we have that µ(t)y (t) + µ (t)y(t) dt = µ(t)y(t). (c) Explain how the above two problems relate to solving first order linear differential equations. Solution: The above two problems outline the derivation of the integrating factor and illustrate how multiplication by the integrating factor makes the left hand side (y + p(x)y) of linear first order differential equations (with a leading coefficient of ) integrable. Furthermore, part b gives the integral explicitly. 3. (problem 5 from B&D) Consider the following differential equation (assume y is a function of t). y y = 3e t () (a) Draw a direction field for equation (). If you choose, you may have a computer do this (using Maple, Matlab, Mathematica, Wolfram Alpha, etc.). Solution: Page 8
9 (b) Based on an inspection of the direction field, describe how solutions behave for large t. Solution: From the direction field, it appears that lim t y(t) = ± depending on the initial condition. (c) Find the general solution of the given differential equation, and use it to determine how solutions behave as t approaches infinity. Solution: The integrating factor µ(t) is e dt = e t Page 9
10 (remember that we can let the constant of integration be zero when finding the integrating factor). Multiplying both sides of the differential equation by µ(t) we have e t y e t y = 3e t Integrating both sides yields e t y = 3e t + c. Remember that the integral of the left hand side is always µ(t)y. The final answer is then y(t) = 3e t + ce t. The limit as t tends to infinity is ± depending on the sign of c. 4. (from problem 6 in B&D) Find the general solution of the following differential equation: ty + y = sin(t), t > 0. Solution: Dividing through by t we have y + y t = sin(t). t We can divide through by t as it is nonzero on the region (0, ). Our integrating factor, µ(t), is then e t dt = e ln(t) = t. We need not worry about the absolute value as t > 0. Integrating both sides, we have t y = t sin(t) dt = t cos(t) cos(t) dt = sin(t) t cos(t) + c. Dividing through by µ(t) gives a final answer of sin(t) t cos(t) t + c t ; t > 0 Page 0
11 5. Find the general solution to the following differential equation: y + t + y = Here are some useful hints and identities for this one. (a) You will probably need to use trig substitution (with t = tan(u)). (b) sec(t) dt = ln( sec(t) + tan(t) ) + C (c) Your final answer should not include the absolute value of anything (be sure to justify why). Solution: Dividing through by + t gives y + y = + t + t Our integrating factor, µ(t), is then ( exp ) dt + t where exp(x) = e x. We then need to find +t dt. Start by letting u = arctan(x). Then tan(u) = x and sec (u) du = dx. It follows that dt = + t sec (u) + tan (u) du = sec(u) du. Integrating sec(u) is such a fun problem that it has its own Wikipedia page. For more fun readers are referred to the Wikipedia page for the integral of sec 3 (x). cos(u) sec(u) du = cos(u) du = cos (u) du = cos(u) sin (u) du Making the substitution v = sin(u) we have dv = cos(u) du and cos(u) sin (u) du = v dv = v + + v dv = ( ln v +ln +v ) = ln + v v = ln + sin(u) sin(u) = ln ( + sin(u)) sin (u) = ln + sin(u) cos(u) ( ) = ln sec(u) + tan(u) = ln + tan (u) + tan(u) = ln + t + t. Page
12 We can ignore the absolute value here as + t + t is positive everywhere. It follows that our integrating factor is µ(t) = exp(ln( + t + t)) = + t + t Integrating both sides of µ(t)y (t) + µ(t)y(t) +t = µ(t) +t ( + t + t)y = + t + t + t dt = dt + we have t + t dt = t + + t + c. To integrate t +t dt we used the substitution u = + t. Dividing through by µ(t) gives a final answer of y(t) = + c t + + t. 6. (B&D #7) Find the solution to this initial value problem. y y = e t, y(0) = Solution: Like problem, the integrating factor µ(t) is e t. Dividing through by µ(t) and integrating we have ye t = dt = t + c. The general solution is then y(t) = te t + ce t. With the initial condition we have y(0) = = 0e 0 + ce 0 = c. The final answer is then y(t) = (t + )e t Page
13 7. (a) (B&D #0) Find the solution to this initial value problem. ty + (t + )y = t, y(ln()) =, t > 0 Solution: Dividing through by t leaves us with y + t + y =. t The integrating factor µ(t) is then ( ) t + µ(t) = exp dt t ( = exp + t dt ) = te t. Dividing through by our integrating factor and integrating both sides yields te t y(t) = te t dt = te t e t dt = (t )e t + c. The general solution is then y(t) = t t Plugging in for the initial condition gives y(ln()) = = ln() ln() + c te t c ln() + c + =, ln()eln() ln() implying that ln() = ln() + c and c =. The final solution then becomes t t + te t. (b) Briefly explain why the condition t > 0 was given. Solution: This condition was given as (0, ) is the largest open interval containing ln() (where the initial value was specified) on which t (the leading coefficient) is nonzero and t+ is continuous. We also use the fact that t was t positive in the above solution when we neglected to use absolute values when integrating. t Page 3
14 8. Optional Hard Problem (B&D #30) Find the value of y 0 for which the solution of the initial value problem remains finite as t. y y = + 3 sin(t), y(0) = y 0 Solution: We will start by solving the differential equation. The integrating factor is µ(t) = e t. Multiplying through by µ(t) and integrating both sides yields ye t = e t + 3e t sin(t) dt = e t + 3 e t sin(t) dt Notice the following. e t sin(t) dt = e t sin(t) e t sin(t) e t cos(t) e t cos(t) dt = e t sin(t) dt implying that e t sin(t) = e t (sin(t) + cos(t)) + c and e t sin(t) = e t (sin(t) + cos(t)) + c. It follows that and ye t = e t 3 e t (sin(t) + cos(t)) + c y(t) = 3 (sin(t) + cos(t)) + cet. Plugging in the initial condition y(0) = y 0 we have y 0 = 3 + c implying that c = y In order for 3 (sin(t) + cos(t)) + cet to stay finite as t, c must be zero. It follows that the value of y 0 that we are looking for is 5. Page 4
15 9. (Problem 8c and c from B&D) For each of the following, solve the differential equations determine its behavior as t. (a) ( + t )y + 4ty = ( + t ) (b) y + y = 5 sin(t) Solution: (a) We begin by dividing everything by + t which gives us the equation y + 4t + t y = ( + t ) 3 () Note that we don t have to worry about the domain of t since + t is never 4t zero. Next, we need to find an integrating factor µ(t) = e +t dt. To solve 4t dt, use u-substitution with u = +t and du = tdt to write the integral +t as du = ln u + C. Thus µ(t) = ( u e ln u +C = C ) e ln u = C u which becomes ( + t ) by choosing C = 0 and noting that u = u for any u. Then equation () becomes ( + t ) y + 4t( + t )y = (3) + t Integrating both sides gives so we have ( + t ) y = = arctan(t) + C (4) + t y = arctan(t) + C ( + t ) (5) Finally, we need to consider the behavior of y as t. However, we know that π < arctan(t) < π so by the Squeeze Theorem, we see π lim + C π t ( + t ) lim arctan(t) + C lim + C t ( + t ) t ( + t ) = 0 lim t arctan(t) + C ( + t ) 0 = lim t arctan(t) + C ( + t ) = 0 Thus, regardless of initial conditions, y 0 as t. Page 5
16 (b) For y + y = 5 sin(t), our integrating factor is just e t so we need to find the integral 5 sin(t)e t dt. We use integration by parts to see 5 sin(t)e t dt = 5 sin(t)e t 0 cos(t)e t dt Using integration by parts again on 0 cos(t)e t dt we get 0 cos(t)e t dt = 0 cos(t)e t 0 sin(t)e t dt Thus we finally have 5 sin(t)e t dt = 5 sin(t)e t 0 cos(t)e t 0 sin(t)e t dt = 5 sin(t)e t dt = 5 sin(t)e t 0 cos(t)e t + C = 5 sin(t)e t dt = sin(t)e t cos(t)e t + C Therefore we have e t y = 5 sin(t)e t dt = sin(t)e t cos(t)e t + C = y = sin(t) cos(t) + Ce t As t gets large, y(t) approaches sin(t) cos(t) since lim t Ce t = 0. Section. 0. (B&D #) Solve the following differential equation. y = x Solution: The above contains a typo and is (as a result) a Calc I problem. It should read y = x y, Page 6
17 in which case we have yy = x. Integrating both sides yields y = 3 x3 + c. Cleaning up a bit and remembering that y cannot be zero, we have a final solution of 3y x 3 = c; y 0.. (B&D #5) Solve the following differential equation. y = ( cos (x) ) ( cos (y) ) Solution: The natural first step in the solution process is to divide through by cos (y), but first we have to consider the case where cos (y) = 0. This happens when y = π + k π. In this case we have a constant solution. 4 Now assume that cos(y) 0. Rearranging the separable equation yields sec (y)y = cos (x). The integral of the left hand side is tan(y). The integral of the right hand side requires just a bit of work. cos (x) dx = cos (x) + dx = cos (x) + (sin (x) + cos (x) dx = (cos (x) sin (x)) + dx = cos(x) + dx = 4 sin(x) + x + c. Equating the two sides and making the solution look all nice and tidy we have tan(x) sin(x) x = c; y π 4 + kπ y = π 4 + kπ ; k Z.. (B&D #9) (a) Find the solution of the following initial value problem in explicit form. y = ( x)y, y(0) = 6 Page 7
18 Solution: Note first that we have an equilibrium solution y(x) = 0. Assuming then that y 0 we can rearrange to yield Integrating both sides we have y y = x. y = x x + c and y = x x + c. Plugging in the initial value we have that c = 6, so the final answer is y = x x + 6. (b) Determine the interval in which the solution is defined. Solution: Note that this problem differs slightly from the one in the book (in the book the initial value given is y(0) = ). In this case, the solution 6 is defined for all x R as x x + 6 has no real roots. It is worthwhile to consider the problem as it is given in the book. 3. (B&D #0) (a) Find the solution of the following initial value problem in explicit form. y ( x ) dy dx = arcsin(x), y(0) = Solution: Assuming x > 0, we have y y = arcsin(x) x. Page 8
19 Integrating both sides (using substitution with u = arcsin(x) on the left hand side) yields 3 y3 = arcsin (x) + c. Plugging in our initial value gives that c = 3 and our final solution becomes 3 y = 3 arcsin(x) +. (b) Determine the interval in which the solution is defined. Solution: The domain of the differential equation was initially restricted by the arcsin(x) on the right hand side and the radical on the right. We are further restricted because dy = 0 when x = ± (consider why this is). Hence dx the interval in which the solution is defined is (, ). 4. (B&D #3) (a) Show that the following differential equation is homogeneous. dy dx = x + 3y xy Solution: Notice that x + 3y xy = ( ) x y + 3y = x ( (y x ) ) y + 3. x (b) Solve the above differential equation. Solution: Making the substitution v = y x The differential equation then becomes Some rearrangement yields we have y = vx and dy dt = x dv dt + v. x dv dt = ( v + 3v ) v = ( v + v ) = + v v. v + v v = x Page 9
20 and we can integrate both sides. ln + v = ln x + c. Exponentiating both sides and noticing that + v > 0 for all v R we have and + v = c x ; c > 0 x + y = cx x. What can we do about those ugly absolute values? Well the original differental equation had issues when x = 0 or y = 0. In particular, we will always need to specify an open interval not containing zero on which our solution is defined. On these intervals x will always have the same sign, so we can absorb it in c. So the final solution becomes x + y = cx 3 ; c > 0, y (Problem from B&D) Solve the separable differential equation y = x y(+x 3 ) Solution: We can separate this as ydy = x + x 3 dx and immediately integrate each side. For the right side, just use u-substitution with u = + x 3, du = 3x to get x + x dx = du 3 3 u = 3 ln u + C = 3 ln + x3 + C Then we have y = 3 ln + x3 + C = y 3 ln + x3 = C 6. (Problem 34 from B&D) Show that the equation dy = 4x+3y dx x+y this property to solve it. is homogeneous and use Page 0
21 Solution: We want to make the change of variables v = y. Dividing the top and x bottom of the right side by x and using the identity dy = v + x dv gives us dx dx v + x dv dx = x dv dx = 4 + 3v + v = 4 + 3v + v v = x dv dx = (4 + 3v) + (v + v ) + v = + v 4 + 5v + v dv = x dx 4 + 5v + v = + v However, we can factor 4 + 5v + v = (v + 4)(v + ) and use partial fractions decomposition. Then A v B v + = + v 4 + 5v + v so we get Av + A + Bv + 4B = + v so Av + Bv = v and A + 4B =. Then solving the second equation for A we get A = 4B so ( 4B)v + Bv = v implying 3B = which in turn implies = 3B. Thus B = so A = 4( ) = Therefore 3(v + 4) + 3(v + ) = + v 4 + 5v + v giving us the differential equation Integrating both sides gives 3 3(v + 4) + 3(v + ) dv = x dx ln v ln v + = ln x + C Putting both sides as the exponent of e, we can reduce to We can then cube both sides to get (v + 4) 3 (v + ) 3 = C x v + 4 (v + ) = C x 3 = C 3 = v + 4 (v + ) x 3 ) ( ) ) = C 3 = ( x v + 4 x v + = ( x(v + 4) ( ) x(v + ) Page
22 Substituting y back in for v gives x C 3 = ( x( yx ) + 4) ( x( yx ) + ) = C 3 = y + 4x y + x Section.3 7. (B&D #) Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains 00 L of a dye solution with a concentration of g/l. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of L/min, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches % of its original value. Solution: Let Q(t) denote the quantity of dye (in grams) in the tank. We know that dq = rate in rate out. dt There is no dye being added to the tank, so the rate in is zero. The rate out is Q(t) g. So we have 00 min dq dt = Q 00 which is both linear and separable. Solving the equation we have Q(t) = ce t 00. From the problem statement we have Q(0) = 00 so c = 00. Determining when the amount of dye in the tank is at % of its initial value amounts to solving 00e t 00 = for t. So the final answer is t = 00 ln(0) min min. 8. (B&D #9) A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 0%. Assuming that interest is compounded continuously and that the borrower makes payments continuously at a constant annual rate k, determine the payment rate k that is required to pay off the loan in 3 years. Also determine how much interest is paid during the 3-year period. Page
23 Solution: The differential equation modeling this situation is dd dt.d = k, where t is in years and D(t) is the debt at time t in dollars. We will solve the linear equation using integrating factor µ(t) = e.t. The general solution is then D(t) = ce t 0 + 0k. Plugging in the initial condition D(0) = 8000 we have c = k and a solution of D(t) = (8000 0k)e t 0 + 0k. In order to determine the annual payment required to pay off the loan in 3 years, we solve for k with the condition D(3) = 0. We have 0 = (8000 0k)e k and Our final solution is then 8000e.3 = 0k(e.3 ). k = 800e.3 e.3 $ The interest paid is 3k 8000 $ (B&D #) Radiocarbon dating is a method of dating organic material by determining the ratio of carbon-4 to other carbon isotopes present in the material. The half-life of carbon-4 is approximately 5730 years (i.e it takes 5730 years for half of the carbon-4 to decay). Let Q(t) be the amount of carbon-4 remaining at time t. (a) Assuming that Q satisfies the differential equation Q decay constant r for carbon-4. = rq, determine the Solution: Solving the equation we have Q(t) = ce rt. From the problem statement we know Q(0) = Q(5730). Solving this we have e r5730 = Page 3
24 and r = ln() 5730 yr. (b) Find an expression for Q(t) at any time t, if Q(0) = Q 0. Solution: Plugging the initial condition in we have Q 0 e ln() 5730 t. 30. Optional Hard Problem Newton s law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and the temperature of its surroundings. Use this law to answer the following question. Two people sitting at a café order coffee. The first person orders creme with the coffee and promptly adds some when the coffee arrives. The second person decides, after a moment of letting the coffee cool, to order creme and adds some when the server brings it. Assuming that the coffees were 00 when they were first brought, the creme is always served at 40, the temperature of the room is 70, and both people add the same quantity of creme to the same quantity of coffee; whose coffee is hotter after person two adds creme? Note that the temperature of the coffee and creme mixture is T o C o + T r C r C o + C r where T o is the temperature of the coffee, T r is the temperature of the creme, C o is the amount of coffee, and C r is the amount of creme. Solution: We begin by modeling the temperature of a cup of coffee with initial temperature T 0. From the problem statement we have that the temperature of the surroundings is 70, so by Newton s law of cooling, dt dt = k(t 70) where T (t) is the temperature of the coffee at time t. Solving the equation we have Plugging in our initial condition yields T (t) = ce kt + 70; c > 0. T (t) = (T 0 70)e kt Page 4
25 We can then model the temperature of person s coffee (after the creme is added) by the equation ( ) 00Co + 40C r T (t) = 70 e kt C o + C r With C o and C r as indicated in the problem statement. Pulling out C o+c r T (t) = Collecting terms we have T (t) = [ (00Co + 40C r 70(C o + C r )) e kt + 70(C o + C r ) ] C o + C r C o + C r [ C0 ( (00 70)e kt + 70 ) + C r ( (40 70)e kt + 70 )]. yields. This indicates that mixing the creme and coffee and letting the mixture cool is equivalent (under our assumptions) to letting the creme heat up and the coffee cool and then mixing them together at time t (the above equation is the result of plugging the equations for temperature of coffee and creme at a given time with initial temperatures 00 /40 into the equation for the temperature of two fluids after mixing). It follows that person s coffee will be warmer because his situation is equivalent to letting the creme heat up and then adding it, while person s creme is kept cold. 3. (Problem 8 from B&D) A college student with no initial capital invests k dollars per year at an annual rate of return r. Assume the investments are made continuously and that the return is compounded continuously. Then (a) Determine the sum S(t) accumulated at any time t (b) If r = 7.5%, determine k so that $ million will be available for retirement in 40 years. (c) If k = $000 per year, determine the return rate r that must be obtained to have $ million available in 40 years. Solution: (a) We see that ds = k + rs = r(s + k ) so we separate the equation and get dt r ds S + k r = rdt Page 5
26 Integrating both sides gives ln S + k r = rt + C = S + k r = C e rt Note that S 0 = S(0) = 0 so = S = C e rt k r 0 = C k r = C = k r Thus S(t) = k r ( ) e rt (b) If r = 7.5% = and S(40) =, 000, 000 we have S(40) =, 000, 000 = k ( ) e 0.075(40).075 = (0.075), 000, 000 e 0.075(40) = k = k $3930 (c) If we now assume k = $000 and S(40) = $, 000, 000, then ( ), 000, 000, 000 = e 40r r = 500r = e 40r Unfortunately, a computer is needed to solve this equation. We get r = = 9.77%. 3. (Problem 5a,c from B&D) A tank contains 00 gal of water and 50 oz of salt. Water containing a salt concentration of ( + oz sin(t)) flows into the tank at a rate of 4 gal, and the mixture in the tank flows out at the same rate. gal min (a) Find the amount of salt Q(t) in the tank at any time. (b) Note that the long-time behavior of the solution is an oscillation about a certain constant level. What is this level? What is the amplitude of the oscillation? Page 6
27 Solution: (a) As usual with tank problems, dq dt = rate in rate out. Here rate in = 4 ( + sin(t)) = ( + 4 sin(t) ) oz min and rate out = Q(t) 00 = Q(t) oz 50 min Therefore we have that dq ( dt = + ) 4 sin(t) Q(t) 50 = dq dt + Q(t) ( 50 = + ) 4 sin(t) This is a first order linear equation so we can solve it with an integrating factor. Here µ(t) = e 50 dt = e t 50. Thus the differential equation becomes Integrating both sides gives We note that to solve t 4 e 50 sin(t)dt. e t dq 50 dt + e t Q(t) = e t 50 + e t 50 4 sin(t) e t 50 Q(t) = e t 50 dt + 4 e t 50 sin(t)dt e t 50 dt = 5e t 50 + C, and proceed to use integration by parts e t 50 sin(t)dt = 50e t 50 sin(t) 50 e t 50 cos(t)dt Integration by parts again gives e t t 50 cos(t)dt = 50e 50 cos(t) + 50 e t 50 sin(t)dt so we finally have e t t 50 sin(t)dt = 50e 50 sin(t) (50) e t 50 cos(t) (50) e t 50 sin(t)dt = ( ) (50) + e t t 50 sin(t) = 50e 50 sin(t)dt (50) e t 50 cos(t) + C Page 7
28 = e t sin(t)dt = ( )e t sin(t) ( )e t 50 cos(t) + C (50) + (50) + At long last we can write e t 50 Q(t) = e t 50 dt + 4 e t 50 sin(t)dt e t t Q(t) = 5e 50 + ( )e t 50 sin(t) ( 50 )e t 50 cos(t) + C 4 (50) (50) = Q(t) = 5 + ( ) sin(t) 4 (50) + which we could reduce to Q(t) = sin(t) However, our initial conditions were Q(0) = 50 so so all told we have 50 = C 50 ( 4 (50) + cos(t) + Ce = C = = Q(t) = sin(t) cos(t) e ) cos(t) + Ce t 50 t 50 t (b) As t, Thus we oscillate around Q(t) = 5. To find the amplitude, we must find the max/min of 50 e t f(t) = 5 65 sin(t) cos(t) To do this, we take the derivative and set it equal to 0. f (t) = 5 65 cos(t) sin(t) 0 = 5 65 cos(t) sin(t) Page 8
29 = 50 sin(t) = cos(t) = tan(t) = 50 Thus f(t) has a min/max at arctan( ). We then have 50 f(arctan( 5 )) = sin(arctan( 65 )) cos(arctan( 50 )) To find sin(arctan( )), suppose u = arctan( ). Draw a right triangle with u as one of the angles. Since u = arctan( ), it follows that tan(u) = 50 so let the leg of the triangle opposite u have length and the other leg 50 have 50. Then the hypotenuse, must have length Thus sin(u) = +50. Similarly, we can see cos(u) = f(arctan( 5 )) = = (500)(50) = so we see the amplitude of the oscilation is Section (Problems and 3 from B&D) For each of the following initial value problems, find an interval of t in which Theorem.4. from the book (Theorem from lecture notes) guarantees the existence of the a (unique) solution. (a) (t 3)y + ln(t)y = t ; y() = (b) y + tan(t)y = sin(t) ; y(π) = 0 Solution: (a) To apply Theorem.4., we need to get the equation into the form dy dt = f(t, y). Thus we consider y = t t 3 ln(t) t 3 Page 9
30 Thus we cannot have t = 3. Furthermore, the presence of ln(t) means we must require t > 0. Therefore we can either choose (0, 3) or (3, ). However, we need the interval to include our initial condition y() =. Thus we choose (0, 3) since it contains t =. (b) We proceed similarly and get y = sin(t) tan(t)y This has a discontinuity at t = (k+)π for k Z since this is exactly where cos(t) = 0 and tan(t) = sin(t). Because our initial condition is y(π) = 0, we cos(t) must choose the interval π < t < 3π. 34. (Problems 7 and 8 from B&D) For each of the following initial value problems, state where in the ty-plane the hypothesis of Theorem.4. from the book (Theorem from lecture notes) are satisfied. (a) y = t y t+5y (b) y = ( t y ) Solution: (a) According to Theorem.4., we must find a rectangle α < t < β, γ < y < δ in which both f and f are continuous. Here y f = t y t + 5y ; f y = 7t (t + 5y) Thus both are discontinuous exactly when t + 5y = 0 or equivalently on the line y = t. Thus our rectangle should either be completely in the region 5 t + 5y > 0 or the region t + 5y < 0. (b) We proceed similarly, noting f = y = ( t y ) ; f y = y ( t y ) Thus we must have that t y > 0 or equivalently, t + y <. Note that this is just the interior of the unit circle in the ty-plane. Page 30
31 35. (Problems 3 and 6 from B&D)For each of the following, solve the initial value problem and determine how the interval in which the solution exists depend on the initial value y 0. (a) y = 4t y ; y(0) = y 0 (b) y = t y(+t 3 ) ; y(0) = y 0 Solution: (a) This equation is separable, so we write it as Integrating both sides gives ydy = 4tdt y = t + C Applying our initial conditions show y 0 = C, giving us the equation y = t + y 0 = y = ± y0 4t Thus we must have t < y 0. Furthermore, since f(t, y) = 4t, we must y have y 0. (b) This equation is also separable, becoming ydy = t + t 3 dt = y = 3 ln( + t3 ) + C Thus applying our initial condition gives C = y 0 (remember ln() = 0). Therefore we have y = ± 3 ln( + t3 ) + y0 Thus we must have 3 ln( + t3 ) + y 0 > 0. (Note that y = 0 is not allowed since our original equation included division by y.) Therefore we have ln( + t 3 ) > 3y 0 = + t 3 > e 3y 0 = t 3 > + e 3y 0 Page 3
32 ( = t > 0 + e 3y Note that in particular this requires t > so + t 3 > 0 as we needed to ensure we were not dividing by 0 or having the argument of ln( + t 3 ) be non-positive. ) Optional Hard Problem Suppose we have a tank of water which starts with an initial concentration of lb of salt per gallon, and initially contains 00 gallons of 0 water. Suppose water with a salt concentration of lb pours into the tank at a rate 5 gal of ( sin(t) cos(t) ) gal. Assume water in the tank is mixed completely and hrs instantaneously. Suppose water drains from the tank at a rate of 00 gal. Let S(t) be hrs the amount of salt in pounds contained in the tank. (a) Construct a differential equation to model this situation. (Hint: The amount of water in the tank is not constant.) (b) Note that the equation is a first order linear equation and find an integrating factor for it. You should explicitly solve the integral involved. (c) Find an expression for S(t). This expression may still contain an integral. Solution: (a) As usual with tank problems, ds = rate in rate out. Here we have dt rate in = ( ) lbs sin(t) cos(t) 5 hr The tricky part is finding the rate out since the amount of water in the tank is not constant. However, we note that the total change in the amount of water in the tank is t ( ) t sin(x) cos(x) 00dx = 00 sin(x) cos(x)dx = 50 sin (t) 0 0 Thus the amount of salt leaving the tank rate out = 00 S(t) sin (t) = 00 S(t) lbs sin (t) hr Page 3
33 Therefore ds dt = ( ) sin(t) cos(t) 5 = ds dt sin (t) S(t) = 5 00 S(t) sin (t) ( sin(t) cos(t) (b) We showed in part (a) that the differential equation was linear and therefore can be solved using an integrating factor. Thus we need to find 00 µ(t) = e sin dt (t) To solve the integral, multiply top and bottom by sec (t) to get 00 sec (t) sin dt = 00 (t) dt = 00 sin (t) sec (t) tan (t) dt Then remember the identity sec (t) = + tan (t) to rewrite the denominator as + tan (t) tan (t) = + tan (t) so we get 00 sec (t) + tan (t) dt From here, it is just a u-substitution problem. It may be useful to remember (or if memory fails, look up) the derivative of arctan(t). Let u = tan(t) meaning du = sec (t). Thus we get 00 Thus the integrating factor is + u du = 00 du + u = 00 arctan(u) = 00 ( ) arctan tan(t) ( ) µ(t) = e 00 arctan tan(t) (c) Now our differential equation becomes µ(t) ds ( ) dt + µ(t) sin (t) S(t) = µ(t) sin(t) cos(t) 5 ) Page 33
34 Integrating both sides gives ( ) e 00 arctan tan(t) S(t) = ( ) ) e 00 arctan tan(t) sin(t) cos(t) dt 5( Thus ( ) S(t) = e 00 arctan tan(t) ( ) ) e 00 arctan tan(t) sin(t) cos(t) dt 5( Page 34
Homework 2 Solutions Math 307 Summer 17
Homework 2 Solutions Math 307 Summer 17 July 8, 2017 Section 2.3 Problem 4. A tank with capacity of 500 gallons originally contains 200 gallons of water with 100 pounds of salt in solution. Water containing
More informationMath 308 Exam I Practice Problems
Math 308 Exam I Practice Problems This review should not be used as your sole source of preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..
More information= 2e t e 2t + ( e 2t )e 3t = 2e t e t = e t. Math 20D Final Review
Math D Final Review. Solve the differential equation in two ways, first using variation of parameters and then using undetermined coefficients: Corresponding homogenous equation: with characteristic equation
More informationHomework Solutions: , plus Substitutions
Homework Solutions: 2.-2.2, plus Substitutions Section 2. I have not included any drawings/direction fields. We can see them using Maple or by hand, so we ll be focusing on getting the analytic solutions
More informationSolutions to the Review Questions
Solutions to the Review Questions Short Answer/True or False. True or False, and explain: (a) If y = y + 2t, then 0 = y + 2t is an equilibrium solution. False: This is an isocline associated with a slope
More informationMath 308 Exam I Practice Problems
Math 308 Exam I Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..
More informationSolutions to the Review Questions
Solutions to the Review Questions Short Answer/True or False. True or False, and explain: (a) If y = y + 2t, then 0 = y + 2t is an equilibrium solution. False: (a) Equilibrium solutions are only defined
More informationFirst Order ODEs, Part II
Craig J. Sutton craig.j.sutton@dartmouth.edu Department of Mathematics Dartmouth College Math 23 Differential Equations Winter 2013 Outline Existence & Uniqueness Theorems 1 Existence & Uniqueness Theorems
More informationName: October 24, 2014 ID Number: Fall Midterm I. Number Total Points Points Obtained Total 40
Math 307O: Introduction to Differential Equations Name: October 24, 204 ID Number: Fall 204 Midterm I Number Total Points Points Obtained 0 2 0 3 0 4 0 Total 40 Instructions.. Show all your work and box
More informationMA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 2 Solutions
MA 214 Calculus IV (Spring 2016) Section 2 Homework Assignment 2 Solutions 1 Boyce and DiPrima, p 60, Problem 2 Solution: Let M(t) be the mass (in grams) of salt in the tank after t minutes The initial-value
More informationMATH 294???? FINAL # 4 294UFQ4.tex Find the general solution y(x) of each of the following ODE's a) y 0 = cosecy
3.1. 1 ST ORDER ODES 1 3.1 1 st Order ODEs MATH 294???? FINAL # 4 294UFQ4.tex 3.1.1 Find the general solution y(x) of each of the following ODE's a) y 0 = cosecy MATH 294 FALL 1990 PRELIM 2 # 4 294FA90P2Q4.tex
More informationMATH 307 Introduction to Differential Equations Autumn 2017 Midterm Exam Monday November
MATH 307 Introduction to Differential Equations Autumn 2017 Midterm Exam Monday November 6 2017 Name: Student ID Number: I understand it is against the rules to cheat or engage in other academic misconduct
More informationMATH 251 Examination I October 10, 2013 FORM A. Name: Student Number: Section:
MATH 251 Examination I October 10, 2013 FORM A Name: Student Number: Section: This exam has 13 questions for a total of 100 points. Show all you your work! In order to obtain full credit for partial credit
More informationP (t) = rp (t) 22, 000, 000 = 20, 000, 000 e 10r = e 10r. ln( ) = 10r 10 ) 10. = r. 10 t. P (30) = 20, 000, 000 e
APPM 360 Week Recitation Solutions September 18 01 1. The population of a country is growing at a rate that is proportional to the population of the country. The population in 1990 was 0 million and in
More informationDifferential equations
Differential equations Math 27 Spring 2008 In-term exam February 5th. Solutions This exam contains fourteen problems numbered through 4. Problems 3 are multiple choice problems, which each count 6% of
More informationCalculus IV - HW 3. Due 7/ Give the general solution to the following differential equations: y = c 1 e 5t + c 2 e 5t. y = c 1 e 2t + c 2 e 4t.
Calculus IV - HW 3 Due 7/13 Section 3.1 1. Give the general solution to the following differential equations: a y 25y = 0 Solution: The characteristic equation is r 2 25 = r 5r + 5. It follows that the
More informationMath 392 Exam 1 Solutions Fall (10 pts) Find the general solution to the differential equation dy dt = 1
Math 392 Exam 1 Solutions Fall 20104 1. (10 pts) Find the general solution to the differential equation = 1 y 2 t + 4ty = 1 t(y 2 + 4y). Hence (y 2 + 4y) = t y3 3 + 2y2 = ln t + c. 2. (8 pts) Perform Euler
More informationGraded and supplementary homework, Math 2584, Section 4, Fall 2017
Graded and supplementary homework, Math 2584, Section 4, Fall 2017 (AB 1) (a) Is y = cos(2x) a solution to the differential equation d2 y + 4y = 0? dx2 (b) Is y = e 2x a solution to the differential equation
More informationSect2.1. Any linear equation:
Sect2.1. Any linear equation: Divide a 0 (t) on both sides a 0 (t) dt +a 1(t)y = g(t) dt + a 1(t) a 0 (t) y = g(t) a 0 (t) Choose p(t) = a 1(t) a 0 (t) Rewrite it in standard form and ḡ(t) = g(t) a 0 (t)
More informationAPPM 2360: Final Exam 10:30am 1:00pm, May 6, 2015.
APPM 23: Final Exam :3am :pm, May, 25. ON THE FRONT OF YOUR BLUEBOOK write: ) your name, 2) your student ID number, 3) lecture section, 4) your instructor s name, and 5) a grading table for eight questions.
More informationThe Fundamental Theorem of Calculus: Suppose f continuous on [a, b]. 1.) If G(x) = x. f(t)dt = F (b) F (a) where F is any antiderivative
1 Calulus pre-requisites you must know. Derivative = slope of tangent line = rate. Integral = area between curve and x-axis (where area can be negative). The Fundamental Theorem of Calculus: Suppose f
More informationHomework Solutions:
Homework Solutions: 1.1-1.3 Section 1.1: 1. Problems 1, 3, 5 In these problems, we want to compare and contrast the direction fields for the given (autonomous) differential equations of the form y = ay
More informationProblem Set. Assignment #1. Math 3350, Spring Feb. 6, 2004 ANSWERS
Problem Set Assignment #1 Math 3350, Spring 2004 Feb. 6, 2004 ANSWERS i Problem 1. [Section 1.4, Problem 4] A rocket is shot straight up. During the initial stages of flight is has acceleration 7t m /s
More information1. Why don t we have to worry about absolute values in the general form for first order differential equations with constant coefficients?
1. Why don t we have to worry about absolute values in the general form for first order differential equations with constant coefficients? Let y = ay b with y(0) = y 0 We can solve this as follows y =
More informationChapter 2 Notes, Kohler & Johnson 2e
Contents 2 First Order Differential Equations 2 2.1 First Order Equations - Existence and Uniqueness Theorems......... 2 2.2 Linear First Order Differential Equations.................... 5 2.2.1 First
More informationCalculus I Review Solutions
Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.
More informationElementary Differential Equations
Elementary Differential Equations George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 310 George Voutsadakis (LSSU) Differential Equations January 2014 1 /
More information20D - Homework Assignment 1
0D - Homework Assignment Brian Bowers (TA for Hui Sun) MATH 0D Homework Assignment October 7, 0. #,,,4,6 Solve the given differential equation. () y = x /y () y = x /y( + x ) () y + y sin x = 0 (4) y =
More informationForm A. 1. Which of the following is a second-order, linear, homogenous differential equation? 2
Form A Math 4 Common Part of Final Exam December 6, 996 INSTRUCTIONS: Please enter your NAME, ID NUMBER, FORM designation, and INDEX NUMBER on your op scan sheet. The index number should be written in
More informationMAT 285: Introduction to Differential Equations. James V. Lambers
MAT 285: Introduction to Differential Equations James V. Lambers April 2, 27 2 Contents Introduction 5. Some Basic Mathematical Models............................. 5.2 Solutions of Some Differential Equations.........................
More informationOrdinary Differential Equations
12/01/2015 Table of contents Second Order Differential Equations Higher Order Differential Equations Series The Laplace Transform System of First Order Linear Differential Equations Nonlinear Differential
More informationMath 266, Midterm Exam 1
Math 266, Midterm Exam 1 February 19th 2016 Name: Ground Rules: 1. Calculator is NOT allowed. 2. Show your work for every problem unless otherwise stated (partial credits are available). 3. You may use
More informationSample Questions, Exam 1 Math 244 Spring 2007
Sample Questions, Exam Math 244 Spring 2007 Remember, on the exam you may use a calculator, but NOT one that can perform symbolic manipulation (remembering derivative and integral formulas are a part of
More informationProblem Sheet 1.1 First order linear equations;
Problem Sheet 1 First order linear equations; In each of Problems 1 through 8 find the solution of the given initial value problem 5 6 7 8 In each of Problems 9 and 10: (a) Let be the value of for which
More informationOrdinary Differential Equations (ODEs)
c01.tex 8/10/2010 22: 55 Page 1 PART A Ordinary Differential Equations (ODEs) Chap. 1 First-Order ODEs Sec. 1.1 Basic Concepts. Modeling To get a good start into this chapter and this section, quickly
More informationMATH 251 Examination I October 8, 2015 FORM A. Name: Student Number: Section:
MATH 251 Examination I October 8, 2015 FORM A Name: Student Number: Section: This exam has 14 questions for a total of 100 points. Show all you your work! In order to obtain full credit for partial credit
More informationMAT 132 Midterm 1 Spring 2017
MAT Midterm Spring 7 Name: ID: Problem 5 6 7 8 Total ( pts) ( pts) ( pts) ( pts) ( pts) ( pts) (5 pts) (5 pts) ( pts) Score Instructions: () Fill in your name and Stony Brook ID number at the top of this
More information3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:
3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable
More informationSolutions to Math 53 Math 53 Practice Final
Solutions to Math 5 Math 5 Practice Final 20 points Consider the initial value problem y t 4yt = te t with y 0 = and y0 = 0 a 8 points Find the Laplace transform of the solution of this IVP b 8 points
More informationFirst Order Differential Equations Chapter 1
First Order Differential Equations Chapter 1 Doreen De Leon Department of Mathematics, California State University, Fresno 1 Differential Equations and Mathematical Models Section 1.1 Definitions: An equation
More informationChapter1. Ordinary Differential Equations
Chapter1. Ordinary Differential Equations In the sciences and engineering, mathematical models are developed to aid in the understanding of physical phenomena. These models often yield an equation that
More informationy0 = F (t0)+c implies C = y0 F (t0) Integral = area between curve and x-axis (where I.e., f(t)dt = F (b) F (a) wheref is any antiderivative 2.
Calulus pre-requisites you must know. Derivative = slope of tangent line = rate. Integral = area between curve and x-axis (where area can be negative). The Fundamental Theorem of Calculus: Suppose f continuous
More informationODE Math 3331 (Summer 2014) June 16, 2014
Page 1 of 12 Please go to the next page... Sample Midterm 1 ODE Math 3331 (Summer 2014) June 16, 2014 50 points 1. Find the solution of the following initial-value problem 1. Solution (S.O.V) dt = ty2,
More information(a) x cos 3x dx We apply integration by parts. Take u = x, so that dv = cos 3x dx, v = 1 sin 3x, du = dx. Thus
Math 128 Midterm Examination 2 October 21, 28 Name 6 problems, 112 (oops) points. Instructions: Show all work partial credit will be given, and Answers without work are worth credit without points. You
More informationOld Math 330 Exams. David M. McClendon. Department of Mathematics Ferris State University
Old Math 330 Exams David M. McClendon Department of Mathematics Ferris State University Last updated to include exams from Fall 07 Contents Contents General information about these exams 3 Exams from Fall
More informationChapter 2: First Order ODE 2.4 Examples of such ODE Mo
Chapter 2: First Order ODE 2.4 Examples of such ODE Models 28 January 2018 First Order ODE Read Only Section! We recall the general form of the First Order DEs (FODE): dy = f (t, y) (1) dt where f (t,
More informationMathematics 3C Summer 3B 2009 Worksheet Solutions NAME: MY WEBSITE: kgracekennedy/sb093c.html
Mathematics 3C Summer 3B 2009 Worksheet Solutions NAME: MY WEBSITE: http://math.ucsb.edu/ kgracekenne/sb093c.html When typing in Latex cutting and pasting code and being distracted by formating lead to
More informationPractice Midterm 1 Solutions Written by Victoria Kala July 10, 2017
Practice Midterm 1 Solutions Written by Victoria Kala July 10, 2017 1. Use the slope field plotter link in Gauchospace to check your solution. 2. (a) Not linear because of the y 2 sin x term (b) Not linear
More informationMath Applied Differential Equations
Math 256 - Applied Differential Equations Notes Existence and Uniqueness The following theorem gives sufficient conditions for the existence and uniqueness of a solution to the IVP for first order nonlinear
More informationC H A P T E R. Introduction 1.1
C H A P T E R Introduction For y > 3/2, the slopes are negative, therefore the solutions are decreasing For y < 3/2, the slopes are positive, hence the solutions are increasing The equilibrium solution
More informationIt is convenient to think that solutions of differential equations consist of a family of functions (just like indefinite integrals ).
Section 1.1 Direction Fields Key Terms/Ideas: Mathematical model Geometric behavior of solutions without solving the model using calculus Graphical description using direction fields Equilibrium solution
More informationMATH 251 Examination I October 5, 2017 FORM A. Name: Student Number: Section:
MATH 251 Examination I October 5, 2017 FORM A Name: Student Number: Section: This exam has 13 questions for a total of 100 points. Show all your work! In order to obtain full credit for partial credit
More information2r 2 e rx 5re rx +3e rx = 0. That is,
Math 4, Exam 1, Solution, Spring 013 Write everything on the blank paper provided. You should KEEP this piece of paper. If possible: turn the problems in order (use as much paper as necessary), use only
More informationMAT 275 Test 1 SOLUTIONS, FORM A
MAT 75 Test SOLUTIONS, FORM A The differential equation xy e x y + y 3 = x 7 is D neither linear nor homogeneous Solution: Linearity is ruinied by the y 3 term; homogeneity is ruined by the x 7 on the
More information2.2 Separable Equations
82 2.2 Separable Equations An equation y = f(x, y) is called separable provided algebraic operations, usually multiplication, division and factorization, allow it to be written in a separable form y =
More informationDifferential Equations Class Notes
Differential Equations Class Notes Dan Wysocki Spring 213 Contents 1 Introduction 2 2 Classification of Differential Equations 6 2.1 Linear vs. Non-Linear.................................. 7 2.2 Seperable
More information1+t 2 (l) y = 2xy 3 (m) x = 2tx + 1 (n) x = 2tx + t (o) y = 1 + y (p) y = ty (q) y =
DIFFERENTIAL EQUATIONS. Solved exercises.. Find the set of all solutions of the following first order differential equations: (a) x = t (b) y = xy (c) x = x (d) x = (e) x = t (f) x = x t (g) x = x log
More informationThree major steps in modeling: Construction of the Model Analysis of the Model Comparison with Experiment or Observation
Section 2.3 Modeling : Key Terms: Three major steps in modeling: Construction of the Model Analysis of the Model Comparison with Experiment or Observation Mixing Problems Population Example Continuous
More informationDecember Exam Summary
December Exam Summary 1 Lines and Distances 1.1 List of Concepts Distance between two numbers on the real number line or on the Cartesian Plane. Increments. If A = (a 1, a 2 ) and B = (b 1, b 2 ), then
More informationFirst-Order ODEs. Chapter Separable Equations. We consider in this chapter differential equations of the form dy (1.1)
Chapter 1 First-Order ODEs We consider in this chapter differential equations of the form dy (1.1) = F (t, y), where F (t, y) is a known smooth function. We wish to solve for y(t). Equation (1.1) is called
More informationLECTURE 4-1 INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS
130 LECTURE 4-1 INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS: A differential equation (DE) is an equation involving an unknown function and one or more of its derivatives. A differential
More informationFirst order differential equations
First order differential equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T. First
More informationLesson 10 MA Nick Egbert
Overview There is no new material for this lesson, we just apply our knowledge from the previous lesson to some (admittedly complicated) word problems. Recall that given a first-order linear differential
More informationMath 122 Fall Handout 15: Review Problems for the Cumulative Final Exam
Math 122 Fall 2008 Handout 15: Review Problems for the Cumulative Final Exam The topics that will be covered on Final Exam are as follows. Integration formulas. U-substitution. Integration by parts. Integration
More informationLecture Notes in Mathematics. Arkansas Tech University Department of Mathematics
Lecture Notes in Mathematics Arkansas Tech University Department of Mathematics Introductory Notes in Ordinary Differential Equations for Physical Sciences and Engineering Marcel B. Finan c All Rights
More informationSolutions of Math 53 Midterm Exam I
Solutions of Math 53 Midterm Exam I Problem 1: (1) [8 points] Draw a direction field for the given differential equation y 0 = t + y. (2) [8 points] Based on the direction field, determine the behavior
More informationFinal Exam SOLUTIONS MAT 131 Fall 2011
1. Compute the following its. (a) Final Exam SOLUTIONS MAT 131 Fall 11 x + 1 x 1 x 1 The numerator is always positive, whereas the denominator is negative for numbers slightly smaller than 1. Also, as
More informationMath 142, Final Exam, Fall 2006, Solutions
Math 4, Final Exam, Fall 6, Solutions There are problems. Each problem is worth points. SHOW your wor. Mae your wor be coherent and clear. Write in complete sentences whenever this is possible. CIRCLE
More informationMath Final Exam Review
Math - Final Exam Review. Find dx x + 6x +. Name: Solution: We complete the square to see if this function has a nice form. Note we have: x + 6x + (x + + dx x + 6x + dx (x + + Note that this looks a lot
More information4 Exact Equations. F x + F. dy dx = 0
Chapter 1: First Order Differential Equations 4 Exact Equations Discussion: The general solution to a first order equation has 1 arbitrary constant. If we solve for that constant, we can write the general
More informationLecture Notes for Math 251: ODE and PDE. Lecture 6: 2.3 Modeling With First Order Equations
Lecture Notes for Math 251: ODE and PDE. Lecture 6: 2.3 Modeling With First Order Equations Shawn D. Ryan Spring 2012 1 Modeling With First Order Equations Last Time: We solved separable ODEs and now we
More informationMATH 162. FINAL EXAM ANSWERS December 17, 2006
MATH 6 FINAL EXAM ANSWERS December 7, 6 Part A. ( points) Find the volume of the solid obtained by rotating about the y-axis the region under the curve y x, for / x. Using the shell method, the radius
More informationBasic Theory of Differential Equations
page 104 104 CHAPTER 1 First-Order Differential Equations 16. The following initial-value problem arises in the analysis of a cable suspended between two fixed points y = 1 a 1 + (y ) 2, y(0) = a, y (0)
More informationMath 226 Calculus Spring 2016 Exam 2V1
Math 6 Calculus Spring 6 Exam V () (35 Points) Evaluate the following integrals. (a) (7 Points) tan 5 (x) sec 3 (x) dx (b) (8 Points) cos 4 (x) dx Math 6 Calculus Spring 6 Exam V () (Continued) Evaluate
More informationAMATH 351 Mar 15, 2013 FINAL REVIEW. Instructor: Jiri Najemnik
AMATH 351 Mar 15, 013 FINAL REVIEW Instructor: Jiri Najemni ABOUT GRADES Scores I have so far will be posted on the website today sorted by the student number HW4 & Exam will be added early next wee Let
More informationMATH 251 Examination I October 9, 2014 FORM A. Name: Student Number: Section:
MATH 251 Examination I October 9, 2014 FORM A Name: Student Number: Section: This exam has 14 questions for a total of 100 points. Show all you your work! In order to obtain full credit for partial credit
More informationMATH 101: PRACTICE MIDTERM 2
MATH : PRACTICE MIDTERM INSTRUCTOR: PROF. DRAGOS GHIOCA March 7, Duration of examination: 7 minutes This examination includes pages and 6 questions. You are responsible for ensuring that your copy of the
More information2t t dt.. So the distance is (t2 +6) 3/2
Math 8, Solutions to Review for the Final Exam Question : The distance is 5 t t + dt To work that out, integrate by parts with u t +, so that t dt du The integral is t t + dt u du u 3/ (t +) 3/ So the
More informationMA Spring 2013 Lecture Topics
LECTURE 1 Chapter 12.1 Coordinate Systems Chapter 12.2 Vectors MA 16200 Spring 2013 Lecture Topics Let a,b,c,d be constants. 1. Describe a right hand rectangular coordinate system. Plot point (a,b,c) inn
More informationDON T PANIC! If you get stuck, take a deep breath and go on to the next question. Come back to the question you left if you have time at the end.
Math 307, Midterm 2 Winter 2013 Name: Instructions. DON T PANIC! If you get stuck, take a deep breath and go on to the next question. Come back to the question you left if you have time at the end. There
More informationChapters 8.1 & 8.2 Practice Problems
EXPECTED SKILLS: Chapters 8.1 & 8. Practice Problems Be able to verify that a given function is a solution to a differential equation. Given a description in words of how some quantity changes in time
More information2.4 Differences Between Linear and Nonlinear Equations 75
.4 Differences Between Linear and Nonlinear Equations 75 fying regions of the ty-plane where solutions exhibit interesting features that merit more detailed analytical or numerical investigation. Graphical
More informationM343 Homework 3 Enrique Areyan May 17, 2013
M343 Homework 3 Enrique Areyan May 17, 013 Section.6 3. Consider the equation: (3x xy + )dx + (6y x + 3)dy = 0. Let M(x, y) = 3x xy + and N(x, y) = 6y x + 3. Since: y = x = N We can conclude that this
More information5.3 Interpretations of the Definite Integral Student Notes
5. Interpretations of the Definite Integral Student Notes The Total Change Theorem: The integral of a rate of change is the total change: a b F This theorem is used in many applications. xdx Fb Fa Example
More informationMath 2214 Solution Test 1D Spring 2015
Math 2214 Solution Test 1D Spring 2015 Problem 1: A 600 gallon open top tank initially holds 300 gallons of fresh water. At t = 0, a brine solution containing 3 lbs of salt per gallon is poured into the
More informationExam 1 Review: Questions and Answers. Part I. Finding solutions of a given differential equation.
Exam 1 Review: Questions and Answers Part I. Finding solutions of a given differential equation. 1. Find the real numbers r such that y = e x is a solution of y y 30y = 0. Answer: r = 6, 5 2. Find the
More information3.1 Derivative Formulas for Powers and Polynomials
3.1 Derivative Formulas for Powers and Polynomials First, recall that a derivative is a function. We worked very hard in 2.2 to interpret the derivative of a function visually. We made the link, in Ex.
More informationWelcome to AP Calculus!!!
Welcome to AP Calculus!!! In preparation for next year, you need to complete this summer packet. This packet reviews & expands upon the concepts you studied in Algebra II and Pre-calculus. Make sure you
More informationChapter 13: Integrals
Chapter : Integrals Chapter Overview: The Integral Calculus is essentially comprised of two operations. Interspersed throughout the chapters of this book has been the first of these operations the derivative.
More informationMATH 104 Practice Problems for Exam 2
. Find the area between: MATH 4 Practice Problems for Exam (a) x =, y = / + x, y = x/ Answer: ln( + ) 4 (b) y = e x, y = xe x, x = Answer: e6 4 7 4 (c) y = x and the x axis, for x 4. x Answer: ln 5. Calculate
More informationHomework 3, due February 4, 2015 Math 307 G, J. Winter 2015
Homework 3, due February 4, 205 Math 307 G, J. Winter 205 In Problems -8, find the general solution of the equation. If initial conditions are given, solve also the initial value problem. is Problem. y
More informationMA 110 Algebra and Trigonometry for Calculus Spring 2017 Exam 3 Tuesday, 11 April Multiple Choice Answers EXAMPLE A B C D E.
MA 110 Algebra and Trigonometry for Calculus Spring 017 Exam 3 Tuesday, 11 April 017 Multiple Choice Answers EXAMPLE A B C D E Question Name: Section: Last digits of student ID #: This exam has twelve
More informationSMA 208: Ordinary differential equations I
SMA 208: Ordinary differential equations I First Order differential equations Lecturer: Dr. Philip Ngare (Contacts: pngare@uonbi.ac.ke, Tue 12-2 PM) School of Mathematics, University of Nairobi Feb 26,
More informationMath 106: Review for Exam II - SOLUTIONS
Math 6: Review for Exam II - SOLUTIONS INTEGRATION TIPS Substitution: usually let u a function that s inside another function, especially if du (possibly off by a multiplying constant) is also present
More informationMath Spring 2014 Homework 2 solution
Math 3-00 Spring 04 Homework solution.3/5 A tank initially contains 0 lb of salt in gal of weater. A salt solution flows into the tank at 3 gal/min and well-stirred out at the same rate. Inflow salt concentration
More informationMath 2930 Worksheet Introduction to Differential Equations
Math 2930 Worksheet Introduction to Differential Equations Week 2 February 1st, 2019 Learning Goals Solve linear first order ODEs analytically. Solve separable first order ODEs analytically. Questions
More information3 Inequalities Absolute Values Inequalities and Intervals... 18
Contents 1 Real Numbers, Exponents, and Radicals 1.1 Rationalizing the Denominator................................... 1. Factoring Polynomials........................................ 1. Algebraic and Fractional
More informationMATH 251 Examination I February 25, 2016 FORM A. Name: Student Number: Section:
MATH 251 Examination I February 25, 2016 FORM A Name: Student Number: Section: This exam has 13 questions for a total of 100 points. Show all your work! In order to obtain full credit for partial credit
More informationMath 116 Second Midterm November 14, 2012
Math 6 Second Midterm November 4, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 8 problems. Note that
More informationCalifornia State University Northridge MATH 280: Applied Differential Equations Midterm Exam 1
California State University Northridge MATH 280: Applied Differential Equations Midterm Exam 1 October 9, 2013. Duration: 75 Minutes. Instructor: Jing Li Student Name: Student number: Take your time to
More information