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Unformatted text preview: Diﬀerential Equations
Homework Assignment 3
Due on Thursday, February 3, 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. Consider the initial value problem
= 2 x2 + y − 2x,
dx y (2) = −4. (a) Show that both y1 (x) = −x2 and y2 (x) = 4 − 4x are solutions of this equation.
Where are these solutions valid?
(b) Explain why the fact that y1 (x) and y2 (x) are distinct solutions of the given initial
value problem does not contradict the uniqueness part of Theorem 2 proved in
(c) Find all the solutions of the given initial value problem that have the form y (x) =
ax + b, where a and b are constants.
Problem 2. Solve the initial value problem
= y + 2t,
dt y (0) = 0. Problem 3. Let φ0 (t) = 0 and use the method of successive approximations to solve the initial
= y + 2t,
dt y (0) = 0. ...
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