# Student name: Earlham College. Fall 2011 December 15, 2011

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1 Student name: Earlham College MATH 320: Differential Equations Final exam - In class part Fall 2011 December 15, 2011 Instructions: This is a regular closed-book test, and is to be taken without the use of notes, books, or other reference materials. This portion of the test adds up to 30 points each question (or its numbered parts) is worth 3 points. (1) Determine whether the following pairs of functions are linearly dependent or independent on the indicated interval (give reasons): (a) f(t) = t, g(t) = 5 t on the interval 1 < t < 1. (b) f(t) = e t, g(t) = 5e t on the interval 1 < t < 1. (2) Let y(t) denote the solution of the initial value problem y 3y 4y = 0, y(0) = 2, y (0) = α Find the value of α if lim t y(t) = 0 (3) Find the general solution of the system [ 2 1 u = 1 2 (4) Discuss what would be a suitable form of a particular solution when using the method of undetermined coefficients to solve y + 4y = sin(2t) + t. Give reasons. (5) If y 1, y 2 are linearly independent solutions of t 3 y + 2ty + te t y = 0, and if W (y 1, y 2 )(2) = 2e, find lim t W (y 1, y 2 )(t). (6) Give brief answers to each of the following. (a) Determine whether the differential equation y = x3 + 1 is exact. x 2 y x2 (b) Determine whether the differential equation in (a) is linear or nonlinear, homogeneous or nonhomogeneous. (c) Give an example of a 1st order linear system of differential equations whose equilibrium point is not at the origin of the phase space. Justify your claim. (d) A simple model for the interaction between two competing species is given by the following system of differential equations dx dy = 3x 2xy, = 2.5y xy dt dt where x(t), y(t) represent the populations. Find the non-zero equilibrium point and determine the trajectory directions in the phase-plane (first quadrant only). ] u End of test

2 Student name: Earlham College MATH 320 : Differential Equations : Fall 2011 Take home portion of the final exam. Pick up from the Science Library, starting sround noon December 12. Complete and return to Science Library within 24 hours after checkout, or before library closing time, whichever comes first. Test must be taken between December (by noon). Instructions: Answer all questions on separate paper not on this sheet! Show all steps. You may use the following reference materials: The textbook, your own class notes and homework, supplementary handouts given in class, any materials posted on the class website that were prepared for this class, and a calculator. Prohibited materials: Any other reference sources, including electronic, printed, written or verbal. This test adds up to 33 points. It consists of questions numbered (1) to (5). (1) [5 pts. each 2] Find the general solution of each of the following (a) y y y + y = 4 sin t (b) (x 2 + y 2 ) dy = 2x(2x + y) dx (2) [5 pts.] Use the method of reduction of order to find the general solution of 2t 2 y + ty 3y = 0 given that one solution is y 1 = 1/t. (3) [6 pts.] Consider the nonlinear 1st order system: [ ] [ d x (x y)(3 x y) = dt y x(1 y) (a) Find all the equilibrium solutions. (b) Linearize the system around each equilibrium solution. (c) Pick any one equilibrium point and classify its type (e.g., node, center, focus, etc.) and determine the solution behavior in its vicinity (stable, unstable). (4) [6 pts.] A mass of 0.25 kg is attached to a spring and causes it to stretch cm. The mass is pulled down 4 cm from its equilibrium position and given an initial downward velocity of 16 cm/sec. (a) Write an initial value problem that models this situation. (b) Solve the IVP you setup in part (a). (c) Find the time t 0 when the mass first returns to its equilibrium position. ] Page 1 of 2

3 (5) [6 pts.] (a) Consider a second order linear homogeneous differential equation of the form P (x)y + Q(x)y + R(x)y = 0 (i) This equation can be made exact by multiplying by an integrating factor µ(x) such that µ(x)p (x)y + µ(x)q(x)y + µ(x)r(x)y = 0 can be written in the form [µ(x)p (x)y ] + [f(x)y] = 0 By equating coefficients in these two equations and eliminating f(x), show that µ must satisfy P µ + (2P Q)µ + (P Q + R)µ = 0 This equation is known as the adjoint of the original equation. (b) An equation of the form (i) is self-adjoint if its adjoint is the same as the original equation. Show that a necessary condition for (i) to be self-adjoint is that P (x) = Q(x). End of test

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### 1. (1 pt) Write the given second order equation as its equivalent system of first order equations. u =

1 (1 pt) Write the given second order equation as its equivalent system of first order equations u + u + 5u = 0 Use v to represent the velocity function, ie v = u (t) Use v and u for the two functions,