# 341hw1 - C that makes the function y = Ce-x satisfy the...

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Homework 1, due Friday September 9th , in class 1. Verify that the function y = 3 e - 2 x is a solution of the diﬀerential equation y 0 + 2 y = 0. 2. Verify that the function y = 17 sin 2 x + 42 cos 2 x is a solution of the diﬀerential equation y 00 + 4 y = 0. 3. Verify that if C 1 and C 2 are any constants, then the function y = C 1 sin 2 x + C 2 cos 2 x is a solution of the diﬀerential equation y 00 + 4 y = 0. 4. Verify that the function y = 1 1 + x 2 is a solution of the diﬀerential equation y 0 + 2 xy 2 = 0. 5a. Verify that any function of the form y = Ce - x is a solution of the diﬀerential equation y 0 + y = 0 . (1) 5b. Find a speciﬁc value of
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Unformatted text preview: C that makes the function y = Ce-x satisfy the diﬀerential equation (1) and also satisﬁes the initial condition y (0) = 17. For question 6, I’m thinking of a function f . The graph of f has the property that at every point, the slope is equal to the sum of x and y . 6a. Express that property as a diﬀerential equation. 6b. If the graph goes through the point (1 , 1), what does the graph look like near that point? 6c. What do you think is the approximate value of f (1 . 001)?...
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