m260hw2 - Differential Equations Homework Assignment 2 Due...

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Unformatted text preview: Differential Equations Homework Assignment 2 Due on Thursday, January 27, at the start of the recitation you are registered in Your homework should have a cover sheet with the following information: course title, recitation, last and first name (as they appear in the roster), number of homework. If you use several sheets, please staple them. The problems should be written neatly and in the order they were assigned. Problem 1. Solve the given initial value problem from scratch (do not use a formula). dy 3 − y = 2t3 e2t , dt t y (1) = 0. Problem 2. Consider the initial value problem dy − 2y = 2t2 + et , dt y (0) = y0 . Find the value of y0 that separates solutions that grow positively as t → ∞ from those that grow negatively. What is the solution that corresponds to this critical value of y0 ? Problem 3. (a) Show that if y1 (x) is the general solution of dy + p(x)y = g1 (x) dx and y2 (x) is the general solution of dy + p(x)y = g2 (x), dx then y (x) = y1 (x) + y2 (x) is the general solution of dy + p(x)y = g1 (x) + g2 (x). dx (b) Use (a) to solve y − 3y = cos x + sin 2x. How do you combine the two constants from the two solutions into one constant? ...
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