MATH 202 Final, Spring 2008

M202 - Differential Equations Sample Quizz 3 Reminder: Final Exam for Sections 5-8 at 8:00 AM on 7th June 2006 in WHCR A series solution or an explicit solution is accepted for full credit if and only if this is indicated in the problem. You may use any method you like to find a solution. For full credit you need to state your result clearly! 1. Solve the differential equation (y′)3 = −x3y/(1 + x2)5). 2. Find a solution y = y(x) of y′ = 1 + √x + y. It suffices to find an implicit solution. 3. Find the general solution y = y(x) of y′′′ + y′ = cos(x) . 4. Find the general solution of y′′ − 4y′ + 3y = 1 1 + e−2x . 5. Find the solution y = y(x) of y′′ + (2 + x + x2)y′ + y = 0 with y(0) = 3 and y′(0) = 5 . It suffices to specify a power series solution here. For full credit, give a complete recursive definition of the coefficients of your series. Compute the first 3 coefficients explicitely. 6. Let (y1, y2) = (y1(t), y2(t)) be the solution of y′ 1 = y1 + y2 + t y′ 2 = −y1 + 2y2 with y1(0) = y2(0) = 1. Compute y2(t) for t ≥ 0. 7. Find the inverse Laplace transforms of s + 5 s(s2 + s + 21) and arctan � 2 s − 7 � . 8. Find the solution Y = Y (t) = (y1(t), y2(t)) of the differential equation Y ′ = � 1 1 0 1 � Y with Y (0) = � 1 1 � For full credit you must find an explicit formula for the coefficients y1(t), y2(t), y3(t). 9. Find the solution Y = Y (t) = (y1(t), y2(t), y3(t)) of the differential equation Y ′ =   0 1 0 −1 0 0 0 0 7   Y with Y (0) =   1 0 1   For full credit you must find an explicit formula for the coefficients y1(t), y2(t), y3(t). 10. Find the eigenvalues and eigenvectors of A = � 1 2 3 2 � What is the general solution of Y ′ = AY , Y = Y (t) = (y1(t), y2(t)). www.amal-aub.org