Due 1/24/12: Section and 6 and the problems below.

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1 Math Homework assignments General instructions: No effort will be made to read illegible homework. In this connection illegible means: unintelligible grammar, handwriting too small or otherwise unreadable, and/or anything else that might offend the delicate eye of your instructor. As a general rule, you should use a separate line for each statement/equation, and not string lots of = signs across the page. Homework is due each Tuesday at the beginning of class. If late homework is accepted not usually!, points will be deducted for lateness. It s likely that only 4-5 problems will be graded carefully from each set; the others will be looked at. Due 1/24/12: Section and 6 and the problems below. 1. A spherical mothball parachlorobenzene at 20 C sublimes from the solid to the gaseous state at a rate of k cc of solid per second per square centimeter of exposed surface. What will be the radius of the mothball at any time t if it was observed to be spherical with a 1/2 cm radius at noon on January 18, and was thereafter kept in a room at 20 C? 2. Carbon-14 dating: 14 C is a radioactive isotope of the common 12 C; it has 8 neutrons and 6 protons, and undergoes β-decay to 14 N with a half-life τ 5700 years. This means that if there s x 14 C at time t, then there s x/2 remaining at time t + τ. 14 C obeys the usual law of radioactive decay. a Find the decay constant k for 14 C. 14 C is naturally created in the upper atmosphere by cosmic rays. It is also unnaturally created by the atmospheric testing of nuclear weapons; even apart from this, there is some natural variation in the rate at which the 14 C is created, but we ll ignore these inconvenient facts in this exercise. The 14 C mixes with the normal 12 C and is incorporated into living bodies, so the fraction of 14 C in living bodies is about the same as that in the atmosphere. When the body dies, however, there is no further influx of 14 C, and the amount of 14 C will decrease according to the law of radioactive decay. Suppose an old log is discovered. It is relatively simple to determine the fraction of the total carbon in the log which is 14 C and compare it to the fraction in living things. For instance, if the log contains only half the 14 C found in living things, then the conclusion is that the tree died about 5700 years ago, assuming that the levels of 14 C in the atmosphere are approximately constant. b The log is found to have only 15% of the 14 C found in living bodies. How long ago did the tree die? Dating things with 14 C is only accurate to about 60,000 years because it only makes up about a trillionth of the naturally occuring carbon, so after 10 half-lives, there s almost nothing left and the experimental errors are too large to give believable results. Geologists use a rather large collection of long-lived radioactive isotopes to date rock formations, track continental drift, and study plate tectonics. 1

2 3. A cup of coffee at 95 C is set out in a room whose temperature is 20 C. After 10 minutes, it has cooled to 80 C. How long will it take to reach a temperature of 30 C? 4. An object of mass m falls in the gravitational field near the earth s surface. In additional to the gravitatational force, there s a force opposing the motion due to air resistance which is proportional to the velocity of the mass. For simplicity, use km as the constant of proportionality for the air resistance. Write down and solve a DE for the velocity of the object. What is it s terminal velocity? Assuming that the object is initially released v0 = 0 at an initial height of h = x0 meters above the surface, find the value of h such that when the object hits the ground, it s moving at 90% of its terminal velocity. Due 1/31/ The logistic equation is originally of the form dn = kn an 2, k, a > 0. The equilibrium points are at N 1 = 0, N 2 = k/a. Let x = N/N 2 and show that, written in terms of x, the logistic equation takes the form = kx1 x. 2. In class, we solved the logistic equation for 0 < x < 1. Write down ALL the solutions, even those in the unphysical region x < Logistic equation with a threshold: = kx α1 x, where 0 < α < 1/3, k > 0. Identify the equilibrium points and classify them as stable or unstable. Solve the equation and find all the physical solutions corresponding to ICs x0 0, Get the java version of dfield running on your computer there s a link on the main page, and use it to get some direction fields and numerical solutions for a few DEs. How can you tell if a direction field corresponds to an autonomous system? Don t hand this stuff in. 5. Using dfield, investigate the behavior of the solutions to the DE, 2x 1 =, t particularly in regard to the existence/uniqueness theorem. Do hand this one in. 2

3 6. Let L = ad 2 + bd + c be a linear differential operator, where a, b, and c may be functions of time and D = d/, D 2 = d 2 / 2. Suppose that x 1 t, x 2 t are two differentiable functions and c 1, c 2 are arbitrary constants. Show that L[c 1 x 1 + c 2 x 2 ] = c 1 L[x 1 ] + c 2 L[x 2 ]. This is the actual meaning of the word linear in this context. 7. Write a program for your calculator that implements Euler s method; you should be able to save the computations in some sort of table so that you can graph the result. Alternatively, you can look for canned programs at the TI site or elsewhere, load them into your calculator, and make the effort to understand them before you try to use them. Try the program out on some simple ODEs that you can solve explicitly, so you can see if you re doing the right thing. When you re satisfied you know what s going on, have a look at the following IVP; dy/ = 1 ty; y0 = 1; Use Eulers method with step size 0.15 and look at the solution on the interval [0, 25]. Now try a step size 0.1 on the same interval. Is there a difference? What s going on? Remark: If you can t get Euler s method running to your satisfaction, don t worry about it. This is more of a discussion problem anyway. Due February 7 Remark: It looks as though we ll be making serious contact with the text beginning in Chapter 6 applications of 2nd order eqns. Soon. 1. Find the general solution using the characteristic equation. Then evaluate c 1, c 2 to satisfy the initial conditions: a ẍ + 3ẋ + 2x = 0; x0 = 1, ẋ0 = 2 b ẍ + 3ẋ 4x = 0; x0 = 1, ẋ0 = 1 2. The unforced, damped linear oscillator obeys the equation mẍ + γẋ + κx = 0; m, γ, κ > 0. Show that if the characteristic equation has two real distinct roots r 1 and r 2, then both roots are negative. What does this mean physically i.e., what happens for large t? 3. Recall that Write out both sides of the equation e iθ = cos θ + i sin θ. e iθ 1+θ 2 = e iθ 1 e iθ 2. Two complex numbers are equal if both their real and imaginary parts are equal. What s your conclusion? 3

4 4. Consider the linear ODE t 2 ẍ + 3tẋ 3x = 0. This does not have constant coefficients, but it can be solved in a similar manner: a Assume there are solutions of the form xt = t r, for some values of r. Substitute this into the ODE and find the two values r 1 and r 2 for which this is true. b Show that xt = c 1 t r 1 + c 2 t r 2 is the general solution to the IVP t 2 ẍ + 3tẋ 3x = 0; xt 0 = x 0, ẋt 0 = ẋ 0 provided that t 0 0. What happens at t = 0? By the way, this ODE is called Euler s DE - here he is again! Due February 14: 1. Problems in your text: Section 5.5 pg 141 1, 2; Section 6.2 pg A 2 nd order DE of the form ẍ = fx can be converted into a first order equation as follows: let y = ẋ. Then d 2 x = dy 2 = dy = y dy. So the original DE becomes y dy = fx, which is separable, and can be solved for y = gx in principle. Then put y = / = gx and solve for x in this new separable equation. Notice that this works for any right hand side fx - in particular, the equation does not have to be linear. a Solve the harmonic oscillator equation in this way. b The equation for an ideal pendulum no friction, massless rigid rod of length L supporting the point mass m is ml d2 θ + mg sin θ = 0. Here θ is the angle made by the rod with the vertical. Solve this for the angular velocity dθ/ using the above technique. You can t do the integral to find θt; it s a special function called an elliptic integral. For small oscillations, since sin θ = θ 1/6θ , we can use the approximation sin θ θ. This linearizes the pendulum equation to give the equation of the harmonic oscillator. 3. a For the DE mẍ + γẋ + κx = e kt, show that the particular solution x p t obtained using the method of undetermined coefficients is x p t = ekt pk, where pr = mr 2 + γr + κ is the characteristic polynomial. 4

5 b Similarly, the particular solution to is given by mẍ + γẋ + κx = e iωt x p t = eiωt piω. Since e iωt = cos t + i sin t, it follows why? that the real and imaginary parts of x p t give the particular solutions to L[x] = cosωt and L[x] = sinωt. Find the real and imaginary parts of x p and thus the two particular solutions. Hint: the simplest thing is to note that 1/piω = A + ib for some A, B, and work out the answer using A and B. Then you can figure out what A and B are using 1/piω. c Engineers write the above equation in the form x p t = Hiωe iωt, where Hiω := 1 piω. H is called the transfer function. It connects the input e iωt to the output x p t. We will encounter H again when we look at Laplace transforms. Recall that the magnitude of a complex number z = x + iy is z = z z = x2 + y 2. The ratio of the magnitude of the output to the magnitude of the input is called the gain and is equal to H. Find the gain in terms of m, γ, κ and ω. d Referring to the previous item, show that the gain is also equal to the amplitude of the solution to L[x] = cosωt. Due February From your textbook: a Section 7.2 pg 194: 1d, 5b, 7f b Section 7.3 pg 203: 1c 2. Consider the DE t 2 ẍ 1+tx = 0. Suppose you ignore the fact that t = 0 is a singular point and attempt to solve the DE using a power series xt = a n t n. What happens? n=0 3. The DE t 3 ẋ 2x = 0 has an irregular singular point at t = 0. Try solving this. What happens? This is the DE satisfied by xt = exp 1 t 2, 5

6 which is a well-known function having the property that a it has derivatives of all orders at t = 0 but b it s power series at t = 0 does not converge to the function xt. Maybe you can use your graphing calculator to figure out why this is so... Due March 6 1. Given the system ẋ 1 = 2x 1 x 2 ẋ 2 = x 1 + 2x 2, or, in vector form = Bx; 2 1 where B = 1 2 a Find a single second order equation for x 2 t, solve it, and then find x 1 t. Then find a second order equation for x 1 t, solve it, and find x 2 t. Up to constant multiples in the obvious places, your solutions should be the same. Don t forget the two constants! b Write the general solution to the system in the form xt = c 1 e t v 1 + c 2 e 3t v 2, where v 1 and v 2 are constant vectors. This is completely determined by what you ve done in the previous part - you don t need to solve for v 1 and v 2. c Referring back to the discussion in class, recall that if y = Ax, and then = Bx, dy = A = ABx = ABA 1 y. Define the matrix A by A = v 1.v 2, where the 2 columns are the two vectors found above. Show that 1 0 ABA 1 =. 0 3 Therefore, in the new coordinates y 1, y 2, the DE takes the form ẏ 1 = y ẏ 2 = 3y, and has the simple solution y 1 = c 1 e t, y 2 = c 2 e 3t. In these coordinates, the system is said to be decoupled. 2. From your text: Section 5.6 variation of parameters pg : 2 and 8. 6

7 Due March Solve the IVP 4 1 = x; x0 = 2 2. Find the general solution to 1 3 = 3 1 x. 3. Find the general solution to 1 1 = 0 1 x. You can t use matrix methods here, but you should write your answer in the vector form x = c 1 x 1 + c 2 x If x 1 t and x 2 t are two solutions to / = Atx on the interval [a, b], and W x 1, x 2 = detx 1.x 2 0 in [a, b], show that c 1 x 1 t + c 2 x 2 t is the general solution to the DE on the interval [a, b]. 5. For the DE in 1 above, write down a fundamental matrix solution Ψt. Find the unique fundamental matrix solution which satisfies φ2 = I. 6. Find a matrix P which diagonalizes the matrix 1 3 A = 3 1 Verify that P 1 AP is diagonal.. Due Tuesday, March Let Ψt = x 1 t.x 2 t be a fundamental matrix solution to the ODE Let φt := ΨtΨ 1 t 0. Show that 2. Let = Ax, where A is constant. φt = e t t 0A. J = Compute the matrix e tj directly, using the definition of the matrix exponential. 7.

8 3. Let Ψt and Ψt be two different fundamental matrix solutions to ẋ = Ax. Show that a There exists a non-singular matrix B such that ΨtB = Ψt, b Therefore ΨtΨ 1 t 0 = Ψt Ψ 1 t 0. c Give an alternative proof of the equality in b using the existence/uniqueness theorem for ODEs. 4. Find the matrix 2 1 e ta, where A = 1 2 by finding the fundamental matrix Φt for the ODE ẋ = Ax. 5. For the logistic equation = kx1 x, 0 < x < 1, with IC x0 = x 0, find the one-parameter group F t, x, and verify explicitly that F t + s, x = F t, F s, x. Solve the IVP above - so the constant of integration should be replaced by an expression involving x 0. Then change x 0 to x; this should be the answer. Due April Computations a Lt 2 = b Lcoshat = Hint: coshat = 1/2e at + e at. c Le 2t cosωt = d Lδt 4 cost 3 t = e L 1 s + 2 = s 2 + 6s + 13 f L 1 s 1 = s 2 + 4s 5 g L 1 se 2s = s 2 + 6s Let gt = Find Lg in terms of Lf. 8 t 0 fu du.

9 3. Page 283 in your text: 11a 4. Differentiating under the integral sign: Suppose we define a function fs by fs = b a gt, s. Then under reasonable circumstances, we can differentiate f with respect to s by differentiating under the integral sign even when b = if g is properly behaved: Reason: We look at f s = b a g t, s. s fs + s fs s = b a gt, s + s gt, s s which leads to the result above as s 0. Using this result, show that F s = L tft, and that F s = Lt 2 ft. Proceeding along these lines, calculate the Laplace transform of t n. 5. Using integration by parts formally find Lδ t a. 6. Write the solution to ẍ + ẋ + x = e t ; x0 = ẋ0 = 0 as a convolution of two functions, and evaluate the integral if you can. Due April Find the Fourier sine series for the function fx = x1 x on the interval [0, 1]. That is, solve for the coefficients a n in the series x1 x = a n sinnπx. n=1 You should be able to find the general term using integration by parts. You should find that the even coefficients vanish, and you should be able to write out the series. And you know what the sum is already! 9

10 2. Using separation of variables, find the general solution as far as we know! to the heat equation on the real line: u t = k 2 u x, k > 0, 2 where the boundary conditions are ut, 0 = ut, L = 0. Here ut, x is the temperature in the homogeneous rod which extends from x = 0 to x = L, and the boundary conditions indicate that the ends of the rod are kept at 0. Then find the specific solution corresponding to the initial condition u0, x = sinπx/l. What happens as t? What does the solution look like for t < 0? Is this physically realistic? Why or why not? Due April The unforced damped pendulum has the equation of motion ml θ + c θ + mg sin θ = 0, c > 0. Rewrite this as a 2-D system and classify all the critical points. Based on this, and your knowledge of its physical behavior, roughly sketch the phase portrait, showing a reasonable selection of orbits in the θ θ plane. Describe what the pendulum is doing on each orbit. 2. Here is another model for competing species: dy = x1 x y = y y 0.75x Based on your analysis of the critical points, discuss the qualitative nature of the solutions. In particular, how, if at all, does this model differ from the one worked out in class? Sketch enough sample orbits to illustrate your description of what s going on. You can check your results afterwards! by plotting some orbits using Polking s pplane app. 10