Make sure this exam has 15 pages. Math 310 Introduction to Ordinary Differential Equations inal Examination August 9, 2006 Instructor: John Stockie Name: (Please Print) Student Number: Special Instructions Q Mark Max 1. ime: 3 hours. 2. No notes, textbooks or calculators are permitted. 3. Before you do anything else, read through all the questions carefully! 4. here is a list of useful formulas at the end of this exam (see page 15). 5. Answer all questions in the space provided. If more space is needed, use the extra pages provided (or the back of the facing page) and clearly indicate where you have continued the question. 6. his exam is intended to test your understanding of concepts, so be sure to explain what you are doing in every step. / 10 1 10 2 10 3 12 4 6 5 6 6 14 7 10 8 12 9 10 otal 100
Math 310 inal Exam Page 2 of 15 rue-alse Questions or each of the questions in this section, circle the letter or, indicating whether you think the answer is true or false. Each question is worth 1 mark, and marks are given only for correct answers. [10 marks] A. Suppose that y 1 (t) and y 2 (t) are two solutions to the equation y + p(t)y + q(t)y = g(t). hen y(t) = 2y 1 (t) y 2 (t) is also a solution. B. here are infinitely many solutions to the differential equation y y = 0 of the form y(x) = ae x, where a is a constant. C. he function y(x) = 1 is the only solution to the IVP 2y y + 2y = 2, y(0) = 1, y (0) = 0, y (0) = 0. D. he following first order equation is both separable and linear: ( x 3 y dy ) = y cos(x 3 ). dx E. he following ODE governing a mass-spring system x + 1 3 x + x = te t has a long-time solution (as t ) that reaches an equilibrium state, where the mass eventually comes to rest.. All solutions of the equation ax + bx + cx = 0 (with a, b and c constants) are of the form e rt, where r is some real number. G. he function f(t) = t log(1 + t) has a Laplace transform. H. he inverse Laplace transform of 3 s 2 + 49 is 3 sin 7t. I. Suppose that A is a constant coefficient, nonsingular, n n matrix, and f is a constant n-vector. hen the linear system of ODE s d y dt = A y + f will always have an equilibrium solution. J. he linear system d y/dt = A y, with A = 1 2 3 3 has one equilibrium point that is a stable spiral.
Math 310 inal Exam Page 3 of 15 Long Answer Section You must provide explanations to accompany your solutions to all problems in this section. Even if an answer is correct, you will receive NO marks unless it is appropriately justified. Show all your work, and if you need extra space, then use the back of the facing page. [10 marks] 1. (a) Show that the following differential equation is not exact: ( sin y 2ye x sin x ) dx dy + cos y + 2e x cosx = 0. (b) Multiply the ODE from 1a by the function e x, and show that the resulting equation is exact. (c) Solve the equation, assuming that y = π when x = 0.
Math 310 inal Exam Page 4 of 15 [10 marks] 2. (a) ind values of the constants a and b such that x 1/2 is an integrating factor for the following ODE: ax b y + y = cosx (b) or the values of a and b from part (a), find the solution y(x) of this equation as an integral, but don t try to evaluate the integral! (c) Circle the plot below which best represents the direction field plot of y(x). Explain your choice. 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0 1 2 3 4 5 x 6 7 8 9 10 0.5 0 1 2 3 4 5 6 7 8 9 10 x 0.5 0 1 2 3 4 5 x 6 7 8 9 10 y(x) y(x) y(x) 1.0 1.0 1.0
Math 310 inal Exam Page 5 of 15 [12 marks] 3. You are given the following ODE: t 2 y 2y = 0. (a) Verify that y 1 (t) = t 2 is a solution to this equation. (b) Use y 1 (t) to find a second solution to the problem (Hint: Use D Alembert s trick, and substitute y(t) = y 1 (t)u(t)). Show that your two solutions form a fundamental set. (c) Use the method of variation of parameters to find the general solution to the following nonhomogeneous problem: t 2 y 2y = 3t 2 1.
Math 310 inal Exam Page 6 of 15
Math 310 inal Exam Page 7 of 15 [6 marks] 4. (a) State the existence uniqueness theorem for a general second order linear initial value problem. (b) Use the theorem to justify whether or not the following problem has a unique solution: t d2 y dt + t dy 2 4 t 2 dt et y = ln t, y(1/4) = 40, y (1/4) = 1, and indicate the largest possible interval on which the solution exists. [6 marks] 5. Write a paragraph to explain the term resonance in the context of periodically forced ODEs, and illustrate using a specific physical example.
Math 310 inal Exam Page 8 of 15 [14 marks] 6. (a) Solve the following linear ODE using the method of undetermined coefficients: y + 3y + 2y = 1 4e t, y(0) = 2, y (0) = 1 (b) Solve the same ODE using the method of Laplace transforms, and verify that your two answers are the same.
Math 310 inal Exam Page 9 of 15
Math 310 inal Exam Page 10 of 15 [10 marks] 7. Consider the following linear system of ODE s: d x dt = 1 2 α 1 x. (a) or what values of α is the equilibrium point (0, 0) a stable node? (b) or the specific value of α = 1 8, determine the general solution by finding the eigenvalues and eigenvectors of the matrix. Draw a plot of the phase plane, showing several representative solution trajectories.
Math 310 inal Exam Page 11 of 15 [10 marks] 8. Consider the following nonlinear system of differential equations: x = 1 x 3xy y = y + 3xy his system is actually a mathematical model for a laser, where y is proportional to the number of photons emitted by the laser, and x is proportional to the number of atoms excited by the photons. (a) ind and classify all equilibrium points for this system. (b) Draw a sketch of the phase plane, taking care to indicate the behaviour of solution curves near each equilibrium point, and the direction of the trajectories for increasing time. (c) What can you say about the long time behaviour (as t ) of the solution (x(t), y(t)) for any initial values (x o, y o ) with x o, y o 0? What does this mean for the output (in photons) of the laser?
Math 310 inal Exam Page 12 of 15
Math 310 inal Exam Page 13 of 15 [10 marks] 9. Answer ONE of the following two questions EIHER 9A OR 9B, but not both. 9A. A 9 volt battery is hooked up to an appliance that behaves as an LC (inductor capacitor) series circuit. he charge on the capacitor in coulombs, q(t), obeys the following differential equation v(t) 9 L d2 q dt 2 + q C = v(t), where L = 1 henry, and C = 0.25 farads. he function v(t) is the voltage supplied by the battery, and initially q(0) = q (0) = 0. 0 10 20 time (hours) Unfortunately, this is not an Energizer brand battery (which we all know keeps going, and going, and going,...), but rather a cheap brand that begins to wear out shortly after it goes into use. A graph of the voltage supplied by the battery is given above, with time t measured in hours. irst, express the voltage applied by the battery in terms of Heaviside step functions, and then solve the resulting initial value problem for the charge q(t) as a function of time. Interpret the results physically. 9B. Solve the following ordinary differential equation by means of a power series about the point x 0 = 0: y xy 2y = 0. Express each of the two linearly independent solutions as an infinite series. Write down the first four non-zero terms in each solution, and give an explicit formula for the coefficients a n in terms of n. inally, find the particular solution that satisfies the initial conditions y(0) = 5, y (0) = 1. t
Math 310 inal Exam Page 14 of 15
Math 310 inal Exam Page 15 of 15 Useful ormulas Laplace transform properties: Definition: L {f(t)} = (s) = 0 e st f(t) dt Linearity: L {c 1 f(t) + c 2 g(t)} = c 1 (s) + c 2 G(s) Derivatives: L {f (t)} = s(s) f(0) L {f (t)} = s 2 (s) sf(0) f (0) L { f (n) (t) } = s n (s) s n 1 f(0) sf (n 2) (0) f (n 1) (0) Heaviside: L {u c (t)f(t c)} = e cs (s) Shift rule: L {e at f(t)} = (s a) s-derivative: L {t n f(t)} = ( 1) n (n) (s) Common Laplace transforms: f(t) (s) f(t) (s) 0 0 t 1 1 s e at 1 s a cosh(bt) sinh(bt) cos(bt) sin(bt) s s 2 b 2 1 s 2 t n n! s n+1, n 0 e at cos(bt) e at sin(bt) s a (s a) 2 + b 2 b (s a) 2 + b 2 b s 2 b 2 e at t n n! (s a) n+1, n 0 s s 2 + b 2 b s 2 + b 2 u c (t) e cs s