N e t I D : N a m e :Midterm#2MATH 286—Differential Equations PlusThursday, March 13• No notes, p ersonal aids or calculator s are permitted.• Answer all questions in the space provided. If you require more space to write your answer, you may continueon the back of the page. There is a blank page at the end of the exam for rough work.• Explain your work! Little or no points will be given for a correct answer with no explanation of h ow you got it.Good luck!Problem 1. (5 points) Consider a homogeneous linear differential equation with consta nt real coefficients whichhas order 6. Suppose y(x) = x2e2xcos(x) is a solution. Write down the general solution.y(x) =Problem 2. (20 points) Find the general solution of x′=1 22 1x.x(t) =Michael Brannan Armin [email protected] [email protected]/4Problem 3. (10 points) The position x(t) of a certain mass on a spring is described by x′′+ cx′+ 5x = F sin (ωt).(a) Assume first that there is no external force, i.e. F = 0. For which values of c is the system overdamped?(b) Now, F0 and the system is undamped, i.e. c = 0. For which values of ω, if any, does resonance occur?Overdamped for: Resonance for:Problem 4. (20 points) Find the general solution of the differential equation y(3)− y = ex+ 7.y(x) =Michael Brannan Armin [email protected] [email protected]/4Problem 5. (20 points) Consider, for x > 0, the second-order differential equationy′′−1 +2xy′+1x+2x2y = 0.(a) Show that the the functions y1(x) = x and y2(x) = x exare solutions to this differential equation.(b) Using the Wronskian, show that y1and y2are linearly independent solutions to the above differential equation.(c) Find, for x > 0, the general solution to the second-order differential equationy′′−1 +2xy′+1x+2x2y = 2x.y(x) =Michael Brannan Armin [email protected] [email protected]/4Problem 6. (20 points) The motio n of a cer tain mass on a spring is described by x′′+ 2x′+ 2x = 5 sin (t).(a) What is the amplitude of the resulting steady periodic oscillations?(b) Assume that the mass is initially at rest (i.e. x(0) = 0, x′(0) = 0) and find the position function x(t).Amplitude: x(t) =Problem 7. (5 points) Let ypbe any solution to the inhomogene ous linear differential equation y′′+ xy = ex. Finda homogeneous linear d iffer ential equation which ypsolves. Hint: Do not attempt to solve the DE.Homogeneous linear DE:Michael Brannan Armin [email protected]
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