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).
− y = 2t3 e2t ,
t y (1) = 0. Problem 2. Consider the initial value problem
− 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
Problem 3. (a) Show that if y1 (x) is the general solution of
+ p(x)y = g1 (x)
and y2 (x) is the general solution of
+ p(x)y = g2 (x),
then y (x) = y1 (x) + y2 (x) is the general solution of
+ p(x)y = g1 (x) + g2 (x).
(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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