1 In-class part. 1.1 Problems. Practice Final, Math 3350

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1 Practice Final, Math In-class part The in-class part of the final is 2 1 hours, and will be roughly the size of this practice test. At least 2 one problem from this practice in-class test will appear on the real final. You can bring one page of notes, and will be given a table of Laplace transforms. On this practice in-class test there are 9 problems worth a total of 85 points and 3 multiplechoice questions worth a total of 15 points. 1.1 Problems 1. (10/100 pts) Solve the equation dy dx = xy2 with initial value y(0) = 2. Check your solution. 2. (10/100 pts) Solve the equation y + 2xy = 1 with initial value y(0) = 1. Check your solution. 3. (10/100 pts) Solve the equation y + 2y = 1 with initial condition y(0) = 1 and y (0) = 1. Check your solution. 4. (10/100 pts) Find the synchronous solution to the equation Check your solution. y + 2y + 6y = 4 cos t. 5. (10/100 pts) Use integration by parts and the definition of the Laplace transform to show that L [f (t)] = s 2 F sf(0) f (0), where F = L [f(t)]. 1

2 6. (10/100 pts) Solve the equation dy dt = y + 2δ(t 3) with initial value y(0) = 1. Check your solution. 7. (10/100 pts) Solve the equation y + 2y + 17y = u(t 1) with initial conditions y(0) = 1, y (0) = 0. Check your solution. 8. (10/100 pts) The equation of motion of a pendulum is d 2 θ dt + γ dθ 2 dt + g L sin θ = 0 where θ is the angle from the vertical, γ is a damping constant, g is the gravitational acceleration, and L is the length of the pendulum. (a) Show that when γ = 0, the energy E = 1 2 ( ) 2 dθ g dt L cos θ is constant. (Hint: compute de, being careful to apply the chain rule correctly). dt (b) Show that when γ > 0, the energy cannot increase. 9. (5/100 pts) Show that π sin(mt) cos(nt) dt = 0 for all m > 0, n > 0. Of what practical use is this fact? π 2

3 1.2 Multiple choice 1. (5/100 pts) Let f and g be functions for which the Laplace transform exists, and let α be a constant. Which of the following are properties of the Laplace transform? (a) L[f] + L[g] = L[f + g] (b) L[f] L[g] = L[f g] (c) αl[f] = L[αf] (d) All of a,b,c. (e) None of a,b,c. (f) Properties a and b but not c. (g) Properties a and c but not b. (h) Properties b and c but not a. 2. (5/100 pts) Suppose you want to find the values of A and B such that (A + 2B) e 2x + (A B) e x = e x for all x. You might try to solve it by equating coefficients of e x and e 2x on both sides of the equation, giving you two equations in two unknowns, Why is this a valid method? (or is it?) A + 2B = 0 A B = 1. (a) Equating coefficients is valid here because e 2x and e x are linearly independent. (b) Equating coefficients is valid here because e 2x and e x are orthogonal. (c) Equating coefficients is not valid in this case, and your calculation is wrong! (d) It s always valid to take subparts of an equation and set them equal to one another, so this calculation does not depend on any special properties of the functions involved. 3. (5/100 pts) Consider the differential equation d 2 y dt + dy 2 dt y = Ay3 + Bsint where A and B are constant parameters. Which of the following is true? (a) The equation is linear, regardless of the values of A and B. (b) The equation is nonlinear, regardless of the values of A and B. (c) The equation is nonlinear unless A = 0. (d) The equation is nonlinear unless B = 0. (e) The equation is nonlinear unless both A = 0 and B = 0. 3

4 2 Take-home part 1. (15/100 pts) Use DSolve to solve the following initial value problems. Use FullSimplify to simplify each solution as much as possible. Plot each solution on the interval t [0, 10]. (a) y + y + 7y = t 2 e t cos 3t with initial conditions y(0) = 0, y (0) = 0. (b) y + ty = 0 with initial conditions y(0) = 1, y (0) = 0. (c) y + 1 t+1 y = te t with initial value y(0) = (10/100 pts) Use Mathematica to compute the partial fraction decomposition of s + 2 (s 2 + 2s + 10) (s + 1). 3. (30/100 pts) Consider the initial value problem y + 2y + 17y = q(t) with initial conditions y(0) = 0 and y (0) = 0, where q(t) is the function 0 t < 1 q(t) = t 2 1 t 2 t 2 e 2 t t > 2. (a) Use Heaviside step functions to write q(t) as an expression without explicit conditionals. (b) Plot the function q(t) on the interval t [0, 10]. (c) Solve the initial value problem using the method of Laplace transforms. (d) Plot the solution on the interval t [0, 10]. Hint: Use Mathematica s Laplace transform and inverse Laplace transform functions. See the Mathematica documentation center for information on these. 4. (30/100 pts) Compute the Fourier coefficients of the periodic function defined by t+π on π t 0 π π t on 0 < t < π π f(t) = f(t 2π) when t π f(t + 2π) when t < π. I suggest doing the integrals using Mathematica s Integrate function. On a single figure, use Mathematica to plot the partial sums of the series with N = 1, 2, 3, 4, and 5 terms. 4

5 5. (15/100 pts) There is an alternate definition of Fourier series in terms of complex exponential functions, f(t) = C m e imt, m= where the coefficients C m are complex numbers. (a) The conjugate z of a complex number z = a + ib is defined to be z = a ib. Show that the conjugate of a complex exponential is given by [ e iθ ] = e iθ. (b) Show that the complex exponentials have the orthogonality property: π e [ { imt e int] 0 m n dt = 2π m = n. π (c) Show that C m = 1 π f(t)e imt dt 2π π 5