ARNOLD PIZER rochester problib from CVS Summer 2003

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1 ARNOLD PIZER rochester problib from CVS Summer 003 Rochester WeBWorK Problem Library WeBWorK assignment DiffEQ due //07 at :00 AM.( pt) setdiffeq/e7.pg Match each of the following differential equations with a solution from the list below.. y 0. y 5y x y 3x y 4. y 5y 6 0 A. e x B. e 3x C. x D. cos x.( pt) setdiffeq/e7 a.pg Match each of the differential equation with its solution.. xy x. y 0 3. y 0y x y 3x y A. e 8x B. 3x x C. x D. sin x 3.( pt) setdiffeq/e7.pg Match each differential equation to a function which is a solution. FUNCTIONS A. 3x x, B. e x, C. sin x, D. x, E. 7exp 4x, DIFFERENTIAL EQUATIONS. xy x. y 0 3. x y 3x y 4. y 6y ( pt) setdiffeq/osu de 3.pg Match the following differential equations with their solutions. The symbols A, B, C in the solutions stand for arbitrary constants. You must get all of the answers correct to receive credit d y dx dx d y dx 6 0 xy x 4y 4 dx dx 8xy 5. x x dx A. Ae 7x Bxe 7x B. Ae 4x C. 3yx 4y 3 C D. Ce 7x 3 E. Acos 4x Bsin 4x 5.( pt) setdiffeq/ur de.pg Just as there are simultaneous algebraic equations (where a pair of numbers have to satisfy a pair of equations) there are systems of differential equations, (where a pair of functions have to satisfy a pair of differential equations). Indicate which pairs of functions satisfy this system. It will take some time to make all of the calculations. y 5 y 3 y y 3 y 5 y A. y e x y 3e x B. y e 4x y e 4x C. y cos x y sin x D. y sin x cos x y cos x sin x E. y e x y e x F. y e x y e x G. y sin x y cos x As you can see, finding all of the solutions, particularly of a system of equations, can be complicated and time consuming. It helps greatly if we stu the structure of the family of solutions to the equations. Then if we find a few solutions we will be able to predict the rest of the solutions using the structure of the family of solutions. 6.( pt) setdiffeq/ur de.pg It can be helpful to classify a differential equation, so that we can predict the techniques that might help us to find a function which solves the equation. Two classifications are the order of the equation (what is the highest number of derivatives involved) and whether or not the equation is linear. Linearity is important because the structure of the the family of solutions to a linear equation is fairly simple. Linear equations can usually be solved completely and explicitly. Determine whether or not each equation is linear:. ty 0. t d y sint t 3. y d y 4. t e t d 4 y 4 d 3 y 3 d y 7.( pt) setdiffeq/e7 4.pg It is easy to check that for any value of c, the function x c x is solution of equation xy 4x x 0. Find the value of c for which the solution satisfies the initial condition y 3 7. c

2 8.( pt) setdiffeq/e7 4a.pg The functions x c x are all solutions of equation: xy 4x x 0 Find the constant c which produces a solution which also satisfies the initial condition y 4 3. c 9.( pt) setdiffeq/e7 3.pg It is easy to check that for any value of c, the function ce x e x is solution of equation y e x. Find the value of c for which the solution satisfies the initial condition y 3 3. c 0.( pt) setdiffeq/e7 3a.pg The family of functions ce x e x is solution of the equation y e x Find the constant c which defines the solution which also satisfies the initial condition y 3. c.( pt) setdiffeq/e7 5.pg Find the two values of k for which y x e kx is a solution of the differential equation y 5y smaller value = larger value =.( pt) setdiffeq/dp7.pg Some curves in the first quadrant have equations A exp 5x where A is a positive constant. Different values of A give different curves. The curves form a family, F. Let P 4 4 Let C be the member of the family F that goes through P. A. Let f x be the equation of C. Find f x. f x B. Find the slope at P of the tangent to C. slope: C. A curve D is perpendicular to C at P. What is the slope of the tangent to D at the point P slope: D. Give a formula g y for the slope at x y of the member of F that goes through x y. The formula should not involve A or x. g y E. A curve which at each of its points is perpendicular to the member of the family F that goes through the point is called an orthogonal trajectory to F. Each orthogonal trajectory to F satisfies the differential equation dx g y where g y is the answer to part D. Find a function h y such that x h y is the equation of the orthogonal trajectory to F that passes through the point P. h y 3.( pt) setdiffeq/dp7.pg The solution of a certain differential equation is of the form y t aexp 3t b exp 8t where a and b are constants. The solution has initial conditions y 0 5 and y 0 Find the solution by using the initial conditions to get linear equations for a and b y t Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

3 ARNOLD PIZER rochester problib from CVS Summer 003 Rochester WeBWorK Problem Library WeBWorK assignment DiffEQDirectionFields due //07 at :00 AM.( pt) setdiffeqdirectionfields/ur de /ur de.pg Match the following equations with their direction field. Clicking on each picture will give you an enlarged view. While you can probably solve this problem by guessing, it is useful to try to predict characteristics of the direction field and then match them to the picture. Here are some han characteristics to start with you will develop more as you practice. A. Set y equal to zero and look at how the derivative behaves along the x axis. B. Do the same for the y axis by setting x equal to 0 C. Consider the curve in the plane defined by setting y =0 this should correspond to the points in the picture where the slope is zero. D. Setting y equal to a constant other than zero gives the curve of points where the slope is that constant. These are called isoclines, and can be used to construct the direction field picture by hand. Go to this page to launch the phase plane plotter to check your answers. (Choose the integral curves utility from the applet menu, enter dx/= to identify the variables x and t and then enter the function you want for /dx = / =...). (You can also login as practice, or practice (use the login name as a password) and you can then practice more versions of this problem and the next one.). x y y. x y y 3. 3sin x A B C.( pt) setdiffeqdirectionfields/ur de /ur de.pg This problem is harder, and doesn t give you clues as to which matches you have right. Stu the previous problem, if you are having trouble. Go to this page to launch the phase plane plotter to check your answers. (You can also login as practice, or practice (use the login name as a password) and you can then practice more versions of this problem and the previous one.) Match the following equations with their direction field. Clicking on each picture will give you an enlarged view.. y 4 y. y 3 6 y x x y 4. xe x y

4 A B C D 3.( pt) setdiffeqdirectionfields/ur de 3.pg Match the following equations with their direction field. Clicking on each picture will give you an enlarged view. While you can probably solve this problem by guessing, it is useful to try to predict characteristics of the direction field and then match them to the picture. Here are some han characteristics to start with you will develop more as you practice. A. Set y equal to zero and look at how the derivative behaves along the x axis. B. Do the same for the y axis by setting x equal to 0

5 C. Consider the curve in the plane defined by setting y =0 this should correspond to the points in the picture where the slope is zero. D. Setting y equal to a constant other than zero gives the curve of points where the slope is that constant. These are called isoclines, and can be used to construct the direction field picture by hand.. y xe x y. x 3cos x 3. sin 3x y 4. y A B C D 4.( pt) setdiffeqdirectionfields/ur de 4.pg Match the following equations with their direction field. Clicking on each picture will give you an enlarged view. While you can probably solve this problem by guessing, it is useful to try to predict characteristics of the direction field and then match them to the picture. Here are some han characteristics to start with you will develop more as you practice. A. Set y equal to zero and look at how the derivative behaves along the x axis. B. Do the same for the y axis by setting x equal to 0 C. Consider the curve in the plane defined by setting y =0 this should correspond to the points in the picture where the slope is zero. D. Setting y equal to a constant other than zero gives the curve of points where the slope is that constant. These are called isoclines, and can be used to construct the direction field picture by hand. 3

6 . sin x. x y 3. x y y 4. e x y y A B C D Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 4

7 ARNOLD PIZER rochester problib from CVS Summer 003 Rochester WeBWorK Problem Library WeBWorK assignment DiffEQ3Separable due /3/07 at :00 AM.( pt) setdiffeq3separable/osu de 3.pg Find the particular solution of the differential equation dx x 6 e y satisfying the initial condition y 6 ln 6. Answer: y= Your answer should be a function of x..( pt) setdiffeq3separable/osu de 3.pg Find the particular solution of the differential equation y 6 dx y satisfying the initial condition y 7. Answer: y= Your answer should be a function of x. x 3.( pt) setdiffeq3separable/jas7 4 5.pg Find u from the differential equation and initial condition. du e 3 6t 3u u 0 u 4.( pt) setdiffeq3separable/jas7 4 5a.pg Solve the separable differential equation for u. du e3u 3t Use the following initial condition: u 0 7 u 5.( pt) setdiffeq3separable/jas7 4 5b.pg Solve the separable differential equation for u. du e3u 4t Use the following initial condition: u 0 u 6.( pt) setdiffeq3separable/ns7 4 0.pg Solve the separable differential equation 8x y x dx 0 Subject to the initial condition: y 0 (function of x only) 7.( pt) setdiffeq3separable/ns7 4 3.pg Find f x if f x satisfies dx 45yx 8 and the y intercept of the curve f x is 5. f x 8.( pt) setdiffeq3separable/ns7 4 3a.pg Find an equation of the curve that satisfies. dx 44yx 0 and whose y intercept is 6. y x (function of x) 9.( pt) setdiffeq3separable/osu de 3 4.pg Find the solution of the differential equation 4 5x 3e 5 x dx y which satisfies the initial condition y 0 y = 0.( pt) setdiffeq3separable/ns7 4 3.pg Find a function y of x such that y x and y 4 (function of x).( pt) setdiffeq3separable/ns7 4 3a.pg Solve the seperable differential equation. 8y x Use the following initial condition: y 8 7 x (function of y).( pt) setdiffeq3separable/ns7 4 8.pg Solve the differential equation y 3 x dx x Use the initial condition y 3 Express y 4 in terms of x y 4 ( function of x) 3.( pt) setdiffeq3separable/ns7 4 8a.pg Solve the seperable differential equation for. dx x xy 3 ;x 0 Use the following initial condition: y 4 y 4 ( function of x) 4.( pt) setdiffeq3separable/ns7 4 8b.pg Find the function y x (for x 0 ) which satisfies the separable differential equation dx 9 3x;x 0 xy with the initial condition: y ( function of x only) 5.( pt) setdiffeq3separable/osu de 3 3.pg Find the solution of the differential equation ln y dx x y which satisfies the initial condition y e y = 6.( pt) setdiffeq3separable/ur de 3.pg A. Solve the following initial value problem: t 4t 95 y with y. (Find y as a function of t.) B. On what interval is the solution valid Answer: It is valid for t.

8 C. Find the limit of the solution as t approaches the left end of the interval. (Your answer should be a number or the word infinite.) Answer C: D. Similar to C, but for the right end. Answer D: 7.( pt) setdiffeq3separable/ur de 3.pg The differential equation dx cos x y 3y 4 6y 39 has an implicit general solution of the form F x y K In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form F x y G x H K Find such a solution and then give the related functions requested. F x y G x H y 8.( pt) setdiffeq3separable/ur de 3 3.pg The differential equation dx 4 x! exp 3y has an implicit general solution of the form F x y K In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form F x y G x H K Find such a solution and then give the related functions requested. F x y G x H y 9.( pt) setdiffeq3separable/ur de 3 4.pg The differential equation dx 56 y! 8 64x y! 8 has an implicit general solution of the form F x y K In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form F x y G x H K Find such a solution and then give the related functions requested. F x y G x H y 0.( pt) setdiffeq3separable/ur de 3 5.pg The differential equation 4 6cos x dx sin x cos y has an implicit general solution of the form F x y K In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form F x y G x H K Find such a solution and then give the related functions requested. F x y G x H y.( pt) setdiffeq3separable/ur de 3 6.pg A.Find y in terms of x if dx x y 7 and y 0 6 y x B. For what x-interval is the solution defined (Your answers should be numbers or plus or minus infinity. For plus infinity enter PINF ; for minus infinity enter MINF.) The solution is defined on the interval: x.( pt) setdiffeq3separable/ur de 3 7.pg The differential equation dx 4x 3 9y 8y has an implicit general solution of the form F x y K In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form F x y G x H K Find such a solution and then give the related functions requested. F x y G x H y 3.( pt) setdiffeq3separable/ur de 3 8.pg The differential equation exp 6x 5 dx 8 sin y 0 cos y has an implicit general solution of the form F x y K In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form F x y G x H K Find such a solution and then give the related functions requested. F x y G x H y 4.( pt) setdiffeq3separable/ur de 3 9.pg A. Solve the following initial value problem: cos t with y tan " (Find y as a function of t.) B. On what interval is the solution valid (Your answer should involve pi.) Answer: It is valid for t.

9 C. Find the limit of the solution as t approaches the left end of the interval. (Your answer should be a number or PINF or MINF. PINF stands for plus infinity and MINF stands for minus infinity.) Answer C: D. Similar to C, but for the right end. Answer D: 5.( pt) setdiffeq3separable/ur de 3 0.pg The differential equation dx 5 7x 45y 63xy has an implicit general solution of the form F x y K In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form F x y G x H K Find such a solution and then give the related functions requested. F x y G x H y Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 3

10 ARNOLD PIZER rochester problib from CVS Summer 003 Rochester WeBWorK Problem Library WeBWorK assignment DiffEQ4LinearstOrder due /4/07 at :00 AM.( pt) setdiffeq4linearstorder/osu de 4 4.pg Find the particular solution of the differential equation 5 4 dx satisfying the initial condition y 0 0. Answer: y= Your answer should be a function of x..( pt) setdiffeq4linearstorder/ur de 4.pg GUESS one function y t which solves the problem below, by determining the general form the function might take and then evaluating some coefficients. 5t Find y t. y t t 3.( pt) setdiffeq4linearstorder/ur de 4.pg GUESS one function y t which solves the problem below, by determining the general form the function might take and then evaluating some coefficients. Find y t. y t exp 5t 4.( pt) setdiffeq4linearstorder/ur de 4 3.pg Find the function satisfying the differential equation t 6e and y 0 0. y 5.( pt) setdiffeq4linearstorder/ur de 4 5.pg Solve the following initial value problem: t 7 6t with y (Find y as a function of t.) 6.( pt) setdiffeq4linearstorder/ur de 4 6.pg Solve the following initial value problem: 0 3 t 4t with y 0 (Find y as a function of t.) 7.( pt) setdiffeq4linearstorder/ur de 4 7.pg Solve the initial value problem 9 t 7 4t for t with y 0 8.( pt) setdiffeq4linearstorder/osu de 4 5.pg Find the particular solution of the differential equation dx ycos x 4cos x satisfying the initial condition y 0 6. Answer: y= Your answer should be a function of x. 9.( pt) setdiffeq4linearstorder/ur de 4 4.pg Solve the initial value problem 9exp t 8exp 5t with y ( pt) setdiffeq4linearstorder/ur de 4 8.pg Solve the initial value problem 45sin t 30cos t with y 0 8.( pt) setdiffeq4linearstorder/ur de 4.pg Solve the following initial value problem: 6 36t with y 0 8 (Find y as a function of t.).( pt) setdiffeq4linearstorder/ur de 4 3.pg Solve the initial value problem 4 sin t for 0 t π and y π$ 9 cost y cos t# sin t 4 3.( pt) setdiffeq4linearstorder/ur de 4 9.pg A. Let g(t) be the solution of the initial value problem t 0 t 0 with g Find g t. g t B. Let f t be the solution of the initial value problem t t with f 0 0 Find f t. f t (Hint: you can try to guess this solution.) C. Find a constant c so that k t f t cg t solves the differential equation in part B and k() = 0.

11 c 4.( pt) setdiffeq4linearstorder/ur de 4 0.pg A. Let g t be the solution of the initial value problem 3 0 with y 0 Find g t. g t B. Let f t be the solution of the initial value problem with y 0 $ 7 Find f t. f t 3 exp 4t C. Find a constant c so that k t f t cg t solves the differential equation in part B and k(0) = 9. c 5.( pt) setdiffeq4linearstorder/ur de 4.pg Find a family of solutions to the differential equation x 3xy dx xd 0 (To enter the answer in the form below you may have to rearrange the equation so that the constant is by itself on one side of the equation.) Then the solution in implicit form is: the set of points (x, y) where F(x,y) = = constant Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

12 ARNOLD PIZER rochester problib from CVS Summer 003 Rochester WeBWorK Problem Library WeBWorK assignment DiffEQ5ModelingWithstOrder due /5/07 at :00 AM.( pt) setdiffeq5modelingwithstorder/ns7 4 3.pg A tank contains 00 L of pure water.a solution that contains 0.0 kg of sugar per liter enters a tank at the rate 3 L/min The solution is mixed and drains from the tank at the same rate. (a) How much sugar is in the tank initially (kg) (b) Find the amount of sugar in the tank after t minutes. amount = (function of t) (c) Find the concentration of sugar in the solution in the tank after 78 minutes. concentration =.( pt) setdiffeq5modelingwithstorder/ns7 4 3a.pg A tank contains 560 L of pure water. A solution that contains 0.06 kg of sugar per liter enters tank at the rate 7 L/min. The solution is mixed and drains from the tank at the same rate. (a) How much sugar is in the tank at the beginning. y 0 (include units) (b) With S representing the amount of sugar (in kg) at time t (in minutes) write a differential equation which models this situation. S f t S Note:. Make sure you use a capital S, ( and don t use S(t), it confuses the computer). Don t enter units for this function. (c) Find the amount of sugar (in kg) after t minutes. S t (function of t) (d) Find the amout of the sugar after 36 minutes. S 36 (include units) Click here for help with units 3.( pt) setdiffeq5modelingwithstorder/ns7 4 3b.pg A tank contains 00 L of pure water. Solution that contains 0.06 kg of sugar per liter enters the tank at the rate 5 L/min, and is thoroughly mixed into it. The new solution drains out of the tank at the same rate. (a) How much sugar is in the tank at the begining y 0 (kg) (b) Find the amount of sugar after t minutes. y t (kg) (Note that this is a function of t) (c) As t becomes large, what value is y t approaching In other words, calculate lim t% y t (kg) 4.( pt) setdiffeq5modelingwithstorder/ns7 4 3c.pg A tank contains 60 kg of salt and 000 L of water.a solution of a concentration 0.05 kg of salt per liter enters a tank at the rate 6 L/min. The solution is mixed and drains from the tank at the same rate. (a) What is the concentration of our solution in the tank initially concentration = (kg/l) (b) Find the amount of salt in the tank after 4 hours. amount = (kg) (c) Find the concentration of salt in the solution in the tank as time approaches infinity. concentration = (kg/l) 5.( pt) setdiffeq5modelingwithstorder/ns7 4 3d.pg A tank contains 80 kg of salt and 000 L of water. Pure water enters a tank at the rate 8 L/min. The solution is mixed and drains from the tank at the rate 4 L/min. (a) What is the amount of salt in the tank initially amount = (kg) (b) Find the amount of salt in the tank after 3.5 hours. amount = (kg) (c) Find the concentration of salt in the solution in the tank as time approaches infinity. (Assume your tank is large enough to hold all the solution.) concentration = (kg/l) 6.( pt) setdiffeq5modelingwithstorder/ns7 4 3e.pg A tank contains 0 L of pure water. A solution that contains 0.09 kg of sugar per liter enters tank at the rate 3 L/min The solution is mixed and drains from the tank at the same rate. (a) How much sugar is in the tank at the beginning. y 0 (include units) (b) Find the amount of sugar (in kg) after t minutes. y t (function of t) (b) Find the amout of the sugar after 90 minutes. y 90 (include units) 7.( pt) setdiffeq5modelingwithstorder/ns7 5.pg A cell of some bacteria divides into two cells every 30 minutes.the initial population is bacteria. (a) Find the size of the population after t hours y t (function of t) (b) Find the size of the population after hours. y (c) When will the population reach 4 T 8.( pt) setdiffeq5modelingwithstorder/ur de 5 7.pg A cell of some bacteria divides into two cells every 30 minutes. The initial population is 400 bacteria. (a) Find the population after t hours y t (function of t) (b) Find the population after 9 hours. y 9 (c) When will the population reach 3600

13 T 9.( pt) setdiffeq5modelingwithstorder/ns7 5 3.pg A bacteria culture starts with 00 bacteria and grows at a rate proportional to its size. After 3 hours there will be 600 bacteria. (a) Express the population after t hours as a function of t. population: (function of t) (b) What will be the population after 9 hours (c) How long will it take for the population to reach ( pt) setdiffeq5modelingwithstorder/ns7 6.pg A population P obeys the logistic model. It satisfies the equation dp P 3 P for P 0 (a) The population is increasing when (b) The population is decreasing when P (c) Assume that P 0 3 Find P 65 P.( pt) setdiffeq5modelingwithstorder/ns7 5 0.pg An unknown radioactive element decays into non-radioactive substances. In 480 days the radioactivity of a sample decreases by 30 percent. (a) What is the half-life of the element half-life: (days) (b) How long will it take for a sample of 00mg to decay to 89 mg time needed: (days).( pt) setdiffeq5modelingwithstorder/ur de 5.pg A bo of mass 7 kg is projected vertically upward with an initial velocity 8 meters per second. The gravitational constant is g 9 8m$ s. The air resistance is equal to k& v& where k is a constant. Find a formula for the velocity at any time ( in terms of k ): v t Find the limit of this velocity for a fixed time t 0 as the air resistance coefficient k goes to 0. (Enter t 0 as t 0.) v t 0 How does this compare with the solution to the equation for velocity when there is no air resistance This illustrates an important fact, related to the fundamental theorem of ODE and called continuous dependence on parameters and initial conditions. What this means is that, for a fixed time, changing the initial conditions slightly, or changing the parameters slightly, only slightly changes the value at time t. The fact that the terminal time t under consideration is a fixed, finite number is important. If you consider infinite t, or the final result you may get very different answers. Consider for example a solution to y =y, whose initial condition is essentially zero, but which might vary a bit positive or negative. If the initial condition is positive the final result is plus infinity, but if the initial condition is negative the final condition is negative infinity. 3.( pt) setdiffeq5modelingwithstorder/ur de 5 8.pg You have 000 dollars in your bank account. Suppose your money is compounded every month at a rate of 0.3 percent per month. (a) How much do you have after t years. y t (function of t) (b) How much do you have after 0 months. y 0 4.( pt) setdiffeq5modelingwithstorder/ur de 5.pg A young person with no initial capital invests k dollars per year in a retirement account at an annual rate of return Assume that investments are made continuously and that the return is compounded continuously. Determine a formula for the sum S t (this will involve the parameter k): S t = What value of k will provide dollars in 4 years k 5.( pt) setdiffeq5modelingwithstorder/ur de 5 3.pg Here is a somewhat realistic example which combines the work on earlier problems. You should use the phase plane plotter to look at some solutions graphically before you start solving this problem and to compare with your analytic answers to help you find errors. You will probably be surprised to find how long it takes to get all of the details of solution of a realistic problem right, even when you know how to do each of the steps. There is partial credit on this problem. There are 840 dollars in the bank account at the beginning of January 990, and money is added and withdrawn from the account at a rate which follows a sinusoidal pattern, peaking in January and in July with money being added at a rate corresponding to 80 dollars per year, while maximum withdrawals take place at the rate of 440 dollars per year in April and October. The interest rate remains constant at the rate of percent per year, compounded continuously. Let y t represents the amount of money at time t (t is in years). y 0 (dollars) Write a formula for the rate of deposits and withdrawals (using the functions sin(), cos() and constants): g t = The interest rate remains constant at percent per year over this period of time. With y representing the amount of money in dollars at time t (in years) write a differential equation which models this situation. f t y'. Note: Use y rather than y t since the latter confuses the computer. Don t enter units for this equation. Find an equation for the amount of money in the account at time t where t is the number of years since January 990. y t (c) Find the amount of money in the bank at the

14 ( beginning of January 000 (0 years later): Find a solution to the equation which does not become infinite (either positive or negative) over time: y t During which months of the year does this non-growing solution have the highest values 6.( pt) setdiffeq5modelingwithstorder/ur de 5 4.pg Newton s law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton s law of cooling. If the coffee has a temperature of 0 degrees Fahrenheit when freshly poured, and minutes later has cooled to 90 degrees in a room at 66 degrees, determine when the coffee reaches a temperature of 55 degrees. The coffee will reach a temperature of 55 degrees in minutes. 7.( pt) setdiffeq5modelingwithstorder/ur de 5 5.pg Susan finds an alien artifact in the desert, where there are temperature variations from a low in the 30s at night to a high in the 00s in the day. She is interested in how the artifact will respond to faster variations in temperature, so she kidnaps the artifact, takes it back to her lab (hotly pursued by the military police who patrol Area 5), and sticks it in an oven that is, a closed box whose temperature she can control precisely. Let T t be the temperature of the artifact. Newton s law of cooling says that T t changes at a rate proportional to the difference between the temperature of the environment and the temperature of the artifact. This says that there is a constant k, not dependent on time, such that T k E T, where E is the temperature of the environment (the oven). Before collecting the artifact from the desert, Susan measured its temperature at a couple of times, and she has determined that for the alien artifact, k Susan preheats her oven to 80 degrees Fahrenheit (she has stubbornly refused to join the metric world). At time t 0 the oven is at exactly 80 degrees and is heating up, and the oven runs through a temperature cycle every π minutes, in which its temperature varies by 0 degrees above and 0 degrees below 80 degrees. Let E t be the temperature of the oven after t minutes. E t At time t 0, when the artifact is at a temperature of 30 degrees, she puts it in the oven. Let T t be the temperature of the artifact at time t. Then T 0 (degrees) Write a differential equation which models the temperature of the artifact. T f t T. Note: Use T rather than T t since the latter confuses the computer. Don t enter units for this equation. 3 Solve the differential equation. To do this, you may find it helpful to know that if a is a constant, then sin t e at a eat asin t cos t T t After Susan puts in the artifact in the oven, the military police break in and take her away. Think about what happens to her artifact as t ) and fill in the following sentence: For large values of t, even though the oven temperature varies between 60 and 00 degrees, the artifact varies from to degrees. (To answer, you will need to use techniques you reviewed in the trig problems on this assignment to assemble two trig functions into one.) 8.( pt) setdiffeq5modelingwithstorder/ur de 5 6.pg Here is a multipart example on finance. Be patient and careful as you work on this problem. You will probably be surprised to find how long it takes to get all of the details of solution of a realistic problem right, even when you know how to do each of the steps. Use the computer to check the steps for you as you go along. There is partial credit on this problem. A recent college graduate borrows dollars at an (annual) interest rate of 8.75 per cent. Anticipating stea salary increases, the buyer expects to make payments at a monthly rate of 950 t$ 0 dollars per month, where t is the number of months since the loan was made. Let y t be the amount of money that the graduate owes t months after the loan is made. y 0 (dollars) With y representing the amount of money in dollars at time t (in months) write a differential equation which models this situation. f t y'. Note: Use y rather than y t since the latter confuses the computer. Don t enter units for this equation. Find an equation for the amount of money owed after t months. y t Next we are going to think about how many months it will take until the loan is paid off. Remember that y t is the amount that is owed after t months. The loan is paid off when y t = Once you have calculated how many months it will take to pay off the loan, give your answer as a decimal, ignoring the fact that in real life there would be a whole number of months. To do this, you should use a graphing calculator or use a plotter on this page to estimate the root. If you use the the xfunctions plotter, then once you have launched xfunctions, pull down the Multigaph Utility from the menu in the upper right hand corner, enter the function you got for y (using x C

15 as the independent variable, sorry!), choose appropriate ranges for the axes, and then eyeball a solution. The loan will be paid off in months. If the borrower wanted the loan to be paid off in exactly 0 years, with the same payment plan as above, how much could be borrowed Borrowed amount = Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 4

16 ARNOLD PIZER rochester problib from CVS Summer 003 Rochester WeBWorK Problem Library WeBWorK assignment DiffEQ6AutonomousStability due /6/07 at :00 AM.( pt) setdiffeq6autonomousstability/ur de 6.pg The graph of the function f x is (the horizontal axis is x.) Given the differential equation x t f x t. List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not these equations are stable, semi-stable, or unstable..( pt) setdiffeq6autonomousstability/ur de 6.pg The graph of the function f x is Given the differential equation x t f x t. List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not these equations are stable, semi-stable, or unstable. 3.( pt) setdiffeq6autonomousstability/ur de 6 3.pg Given the differential equation x * x 3 5,+- x 5 3 x 0 5 x 5. List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not these equations are stable, semi-stable, or unstable. (It helps to sketch the graph. xfunctions will plot functions as well as phase planes. ) 4.( pt) setdiffeq6autonomousstability/ur de 6 4.pg Given the differential equation x t, x 4 3x 3 8x x 6. List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not these equations are stable, semi-stable, or unstable. (It helps to sketch the graph. xfunctions will plot functions as well as phase planes. ) (the horizontal axis is x.)

17 Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

18 ARNOLD PIZER rochester problib from CVS Summer 003 Rochester WeBWorK Problem Library WeBWorK assignment DiffEQ7Exact due /7/07 at :00 AM.( pt) setdiffeq7exact/ur de 7.pg The following differential equation is exact. Find a function F(x,y) whose level curves are solutions to the differential equation F(x,y) = y xdx 0.( pt) setdiffeq7exact/ur de 7.pg Use the mixed partials check to see if the following differential equation is exact. If it is exact find a function F(x,y) whose level curves are solutions to the differential equation 4x 4 3y dx 3x y d 0 F(x,y) = 3.( pt) setdiffeq7exact/ur de 7 3.pg Use the mixed partials check to see if the following differential equation is exact. If it is exact find a function F(x,y) whose level curves are solutions to the differential equation 3xy 3x y x d 0 F(x,y) = y dx 4.( pt) setdiffeq7exact/ur de 7 4.pg Use the mixed partials check to see if the following differential equation is exact. If it is exact find a function F(x,y) whose level curves are solutions to the differential equation F(x,y) = 3x 3 y dx x 3y 5.( pt) setdiffeq7exact/ur de 7 5.pg Use the mixed partials check to see if the following differential equation is exact. If it is exact find a function F(x,y) whose level curves are solutions to the differential equation e x sin y 3y dx 3x e x cos y d 0 F(x,y) = 6.( pt) setdiffeq7exact/ur de 7 6.pg Check that the equation below is not exact but becomes exact when multiplied by the integrating factor. x y 3 x y 0 Integrating factor: µ x y $. xy 3. Solve the differential equation. You can define the solution curve implicitly by a function in the form F x y G x H K F(x,y) = 7.( pt) setdiffeq7exact/ur de 7 7.pg Find an explicit or implicit solutions to the differential equation F(x,y) = x xy dx xd 0 Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

19 ARNOLD PIZER rochester problib from CVS Summer 003 Rochester WeBWorK Problem Library WeBWorK assignment DiffEQ8FundTheorem due /8/07 at :00 AM.( pt) setdiffeq8fundtheorem/ur de 8.pg This problem involves using the uniqueness property (from the Fundamental Theorem of ordinary differential equations.) It can t be graded by WeBWorK, but is to be handed in at the first class after the due date. A. State the uniqueness property of the fundamental theorem. B. Show directly using the differential equation, that if y t is a solution to the differential equation y t/ y t, then y t' y t a is also a solution to the differential equation. (You will need to use the known facts about y to calculate that y t y t ). (We know that the solution is the exponential function, but you will not need to use this fact.) C. Describe the relationship between the graphs of y and y and using a sketch of the direction field explain why it is obvious that if y is a solution then y has to be a solution also. D. Describe in words why if y t is any solution to the differential equation f y then y t y t a is also a solution. E. Show that if y t solves y t y t, then y t Ay t also solves the same equation. F. Suppose that y t solves y t y t and y 0. (Such a solution is guaranteed by the fundamental theorem.). Let y t y t a and let y 3 t y a y t. Calculate the values y 0 and y 3 0. Use the uniqueness property to show that y t y 3 t for all t. G. Explain how this proves that any solution to y must be a function which obeys the law of exponents. H. Let z x iy. Define exp z ( or e z ) using a Taylor series. Show that if z x iy is a constant, then by differentiating the power series. d exp tz zexp tz I. Use your earlier results to show that exp z w0 exp z exp w. This method of checking the law of exponents is MUCH easier than expanding the power series. You can find a direction field plotter here or at the direction field plotter page. Choose integral curves utility from the main screen menu of xfunctions to get to the phaseplane plotter..( pt) setdiffeq8fundtheorem/ur de 8.pg This problem involves using the uniqueness property (from the Fundamental Theorem of ordinary differential equations.) It can t be graded by WeBWorK, but is to be handed in at the first class after the due date. A. Using the same technique as in the previous problem show that if a function y t satisfies: () y 0 = and () y t' y t then y t r y rt B. Explain in words how this relates to another law of exponents. You can find a direction field plotter htmllink( here ) or at the htmllink( direction field plotter page ). Choose integral curves utility from the main screen menu of xfunctions to get to the phaseplane plotter.

20 3.( pt) setdiffeq8fundtheorem/ur de 8 3.pg This problem involves using the uniqueness property (from the Fundamental Theorem of ordinary differential equations.) It can t be graded by WeBWorK, but is to be handed in at the first class after the due date. Use the same ideas as in the previous problems. A. Suppose that y t satisfies the equation y 0 and y 0 0and y 03. Such a function exists because of the fundamental theorem. (We all know that it is sin t, but you should not use that fact in answering the questions below.) Show that y t y t also satisfies the equation y 0 and that y 0 and y 0 0. B. If y 3 t y t show, using the uniqueness property, that y 3 t C. State the uniqueness property for solutions to second order differential equations (or equivalently to a system of two first order differential equations). D. Use the uniqueness property to show that y t a y a y t y t y a y t y a y t y a y t The formulas for the sin of sums of angles can be calculated completely from the one fact that it satisfies a differential equation. This is a general fact. Any solution of a differential equation has the potential for obeying certain laws which are dictated by the differential equation. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

21 ARNOLD PIZER rochester problib from CVS Summer 003 Rochester WeBWorK Problem Library WeBWorK assignment DiffEQ9LinearndOrderHomog due /9/07 at :00 AM.( pt) setdiffeq9linearndorderhomog/ur de 9.pg Find y as a function of t if 400y 8 0 y 0 y 0 6 y t.( pt) setdiffeq9linearndorderhomog/ur de 9 6.pg Find y as a function of t if 0000y 8 0 with y 0 9 y ( pt) setdiffeq9linearndorderhomog/ur de 9 8.pg Find y as a function of t if 65y 79 0 y 0 4 y ( pt) setdiffeq9linearndorderhomog/ur de 9.pg Find y as a function of t if y 3y 8 0 y 0 5 y 4 y t Remark: The initial conditions involve values at two points. 5.( pt) setdiffeq9linearndorderhomog/ur de 9 7.pg Find y as a function of t if 9y 4y 49 0 y 0 y ( pt) setdiffeq9linearndorderhomog/ur de 9 3.pg Find y as a function of t if 6y 3 0 y 0 y 0 9 y t Note: This particular webwork problem can t handle complex numbers, so write your answer in terms of sines and cosines, rather than using e to a complex power. 7.( pt) setdiffeq9linearndorderhomog/ur de 9 4.pg Find y as a function of t if 64y 44y 87 0 y 0 4 y 0 9 Note: This problem cannot interpret complex numbers. You may need to simplify your answer before submitting it. 8.( pt) setdiffeq9linearndorderhomog/ur de 9 5.pg Find y as a function of t if 40y y 4 0 y 0 y 0 3 y t Note: This problem cannot interpret complex numbers. You may need to simplify your answer before submitting it. 9.( pt) setdiffeq9linearndorderhomog/ur de 9.pg Find y as a function of t if y 6y 45 0 y 0 8 y 0 5 Note: This problem cannot interpret complex numbers. You may need to simplify your answer before submitting it. 0.( pt) setdiffeq9linearndorderhomog/ur de 9 9.pg Find y as a function of t if 6y 8y 0 y 0 6 y 0 Note: This problem cannot interpret complex numbers. You may need to simplify your answer before submitting it..( pt) setdiffeq9linearndorderhomog/ur de 9 0.pg " Find the function y of t which is the solution of 5y 0y 0 with initial conditions y 0 y 0 0 y Find the function y of t which is the solution of 5y 0y 0 with initial conditions y 0 0 y 0 y Find the Wronskian W t W y y " W t Remark: You can find W by direct computation and use Abel s theorem as a check. You should find that W is not zero and so y and y form a fundamental set of solutions of 5y 0y 4 0.( pt) setdiffeq9linearndorderhomog/ur de 9.pg Find y as a function of t if 36y 56y 69 0 y 0 8 y ( pt) setdiffeq9linearndorderhomog/ur de 9 3.pg Find y as a function of t if 4y y 9 0 y 3 4 y ( pt) setdiffeq9linearndorderhomog/ur de 9 4.pg Determine whether the following pairs of functions are linearly independent or not.. f θ' 3cos3θ and g θ 5cos 3 θ 39cosθ. f t' t and g t & t& 3. f x e 3x and g x e 3 x

22 5.( pt) setdiffeq9linearndorderhomog/ur de 9 5.pg Suppose that the Wronskian of two functions f t and f t is given by W t t 4 det 5 f t f t f t f t76 Even though you don t know the functions f and f you can determine whether the following questions are true or false.. The vectors f 4 f 4 and f 4 f 4 are linearly independent. The equations a f a f b f 8 c b f 9 d have a unique solution for any c and d 3. The vectors f f are linearly independent 4. The equations a f a f and f b f : 0 b f : 0 f have more than one solution. 5. The vectors f 0 f 0 and f 0 f 0 are linearly independent 6.( pt) setdiffeq9linearndorderhomog/ur de 9 6.pg Determine which of the following pairs of functions are linearly independent.. f θ' cos 3θ g θ cos 3 θ cos θ. f x e x g x e x 3 3. f x x 3 g x & x& 3 4. f t' 3t g t ;& t& 7.( pt) setdiffeq9linearndorderhomog/ur de 9 7.pg Match the second order linear equations with the Wronskian of (one of) their fundamental solution sets.. y cos t y 0. y y t 0 3. y t y 0 t 0 4. y ty 0 5. y ln t y 0 A. W t t B. W t 3exp t C. W t 7t D. W t esin t E. W t e t ln t t 8.( pt) setdiffeq9linearndorderhomog/ur de 9 8.pg Find y as a function of x if y x y 3xy 4 0 y 8 9.( pt) setdiffeq9linearndorderhomog/ur de 9 9.pg Find y as a function of x if y x y 7xy y Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

23 ARNOLD PIZER rochester problib from CVS Summer 003 Rochester WeBWorK Problem Library WeBWorK assignment DiffEQ0LinearndOrderNonhom due /0/07 at :00 AM.( pt) setdiffeq0linearndordernonhom/ur de 0 4.pg Find a single solution of y if.( pt) setdiffeq0linearndordernonhom/ur de 0 5.pg Use the method of undetermined coefficients to find one solution of y 5y exp 7<+ t. (It doesn t matter which specific solution you find for this problem.) 3.( pt) setdiffeq0linearndordernonhom/ur de 0 3.pg There is an error in this problem it has been marked correct for everyone. I ll get a replacement for it rea for the next problem set. Sorry about that. Take care, Mike Use the method of undetermined coefficients to find one solution of y y 6 7t 6t exp 4t. Note that the method finds a specific solution, not the general one. 4.( pt) setdiffeq0linearndordernonhom/ur de 0 6.pg Use the method of undetermined coefficients to find one solution of y 4 4y 54 6exp 7t cos 3t 48exp 7t sin 3t 6 + exp <+ t. (It doesn t matter which specific solution you find for this problem.) 5.( pt) setdiffeq0linearndordernonhom/ur de 0 7.pg Use the method of undetermined coefficients to find one solution of y y = 0t 7 exp t cos t t 5 exp t sin t. (It doesn t matter which specific solution you find for this problem.) 6.( pt) setdiffeq0linearndordernonhom/ur de 0.pg Find a particular solution to the differential equation y 6y 5 50t 3 y p 7.( pt) setdiffeq0linearndordernonhom/ur de 0 0.pg Find a particular solution to y y 5 5e t. y p y p y p 8.( pt) setdiffeq0linearndordernonhom/ur de 0 8.pg Find a particular solution to the differential equation y 0y t 4t t 6e y p 9.( pt) setdiffeq0linearndordernonhom/ur de 0.pg Find a particular solution to y 6y 8 0te t. 0.( pt) setdiffeq0linearndordernonhom/ur de 0 9.pg Find a particular solution to y 4 6sin t..( pt) setdiffeq0linearndordernonhom/ur de 0.pg Find the solution of y 4y 98exp 5 t with y 0 and y 0.( pt) setdiffeq0linearndordernonhom/ur de 0.pg Find the solution of y y 98exp 8 t with y 0 6 and y ( pt) setdiffeq0linearndordernonhom/ur de 0 3.pg Find the solution of y 7 39 sin 7t 39 cos 7t with y 0 4 and y ( pt) setdiffeq0linearndordernonhom/ur de 0 4.pg Find y as a function of x if x y xy 54 x 3 y 6 y 4 5.( pt) setdiffeq0linearndordernonhom/ur de 0 5.pg Find y as a function of x if x y 9xy 6 x 9 y 5 y 5

24 Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

25 ARNOLD PIZER rochester problib from CVS Summer 003 Rochester WeBWorK Problem Library WeBWorK assignment DiffEQModelingWithndOrder due //07 at :00 AM.( pt) setdiffeqmodelingwithndorder/ur de.pg Another realistic problem: The following problem is similar to the problem in an earlier assignment about the bank account growing with periodic deposits. The basic procedure for this problem is not too hard, but getting details of the calculation correct is NOT easy, and may take some time. A ping-pong ball is caught in a vertical plexiglass column in which the air flow alternates sinusoidally with a period of 60 seconds. The air flow starts with a maximum upward flow at the rate of 6 m$ s and at t 30 seconds the flow has a minimum (upward) flow of rate of 5 7m$ s. (To make this clear: a flow of 5m$ s upward is the same as a flow downward of 5m$ s. The ping-pong ball is subjected to the forces of gravity ( mg) where g 9 8m$ s and forces due to air resistance which are equal to k times the apparent velocity of the ball through the air. What is the average velocity of the air flow You can average the velocity over one period or over a very long time the answer should come out about the same right. (Include units). Write a formula for the velocity of the air flow as a function of time. A t Write the differential equation satisfied by the velocity of the ping-pong ball (relative to the fixed frame of the plexiglass tube.) The formulas should not have units entered, but use units to trouble shoot your answers. Your answer can include the parameters m - the mass of the ball and k the coefficient of air resistance, as well as time t and the velocity of the ball v. (Use just v, not v(t) the latter confuses the computer.) v t Use the method of undetermined coefficients to find one periodic solution to this equation: v t = Calculate the specific solution that has initial conditions t 0 and w w t Think about what effect increasing the mass has on the amplitude, on the phase shift Does this correspond with your expectations.( pt) setdiffeqmodelingwithndorder/ur de.pg A steel ball weighing 8 pounds is suspended from a spring. This stretches the spring 45 8 feet. The ball is started in motion from the equilibrium position with a downward velocity of 6 feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second). Suppose that after t seconds the ball is y feet below its rest position. Find y in terms of t. (Note that this means that the postiive direction for y is down.) Take as the gravitational acceleration 3 feet per second per second. 3.( pt) setdiffeqmodelingwithndorder/ur de 3.pg A hollow steel ball weighing 4 pounds is suspended from a spring. This stretches the spring feet. The ball is started in motion from the equilibrium position with a downward velocity of 8 feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second). Suppose that after t seconds the ball is y feet below its rest position. Find y in terms of t. (Note that the positive direction is down.) Take as the gravitational acceleration 3 feet per second per second. Find the amplitude and phase shift of this solution. You do not need to enter units. v t cos + t Find the general solution, by adding on a solution to the homogeneous equation. Notice that all of these solutions tend towards the periodically oscillating solution. This is a generalization of the notion of stability that we found in autonomous differential equations. 4.( pt) setdiffeqmodelingwithndorder/ur de 4.pg This problem is an example of critically damped harmonic motion. A hollow steel ball weighing 4 pounds is suspended from a spring. This stretches the spring 8 feet. The ball is started in motion from the equilibrium position with a downward velocity of 6 feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second). Suppose that after t seconds the ball is y feet below its rest position. Find y in terms of t.