math54 - quiz 9 - solns - Math 54 quiz 9 Solutions 1. Find...

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Unformatted text preview: Math 54 quiz 9 Solutions 1. Find the general solution to the differential equation y + 2y + ky = 0, where (a) k = 0, (b) k = 1, and (c) k = 2. (a) The roots of the characteristic equation are 0, and -2, so the solution is y (x) = c1 + c2 e−2x . (b) The root −1 is a repeated root, so the solution is y (x) = c1 e−x + c2 xe−x . (c) The (complex) roots are −1 ± i so the solution is y (x) = e−x (c1 sin(x) + c2 cos(x)). 2. Suppose that y1 satisfies the differential equation x2 y + sin(x)y = f (x), and that y2 satisfies the differential equation x2 y + sin(x)y = g (x). Show that c1 y1 + c2 y2 satisfies x2 y + sin(x)y = c1 f (x) + c2 g (x). In order to show that c1 y1 + c2 y2 satisfies the differential equation, we need to plug it in to the left hand side and see what we get: substituting c1 y1 + c2 y2 in for y , we obtain x2 (c1 y1 +c2 y2 ) +sin x(c1 y1 +c2 y2 ) = c1 x2 y1 +c2 x2 y2 +c1 sin(x)y1 +c2 sin(x)y2 = c1 x2 y1 + sin(x)y1 + and recalling that y1 and y2 satisfy the differential equations given above, this is c1 f (x) + c2 g (x), as required. 1 3. Find the general solution to the differential equation y − y − 2y = e2x We use undetermined coefficients. The roots of the characteristic equation are −2, +1, so the homogeneous solution is yH (x) = c1 e−x + c2 e2x . The right side is e2x , so our first guess is Ae2x , but this is already the second solution to the homogeneous equation, so we change the guess to Axe2x . We take two derivatives of this guess to get Ae2x + 2Axe2x and 4Ae2x + 4Axe2x , respectively. Thus, this guess plugged into the left hand side of the equation is 4Ae2x +4Axe2x −Ae2x −2Axe2x −2Axe2x = xe2x (4A−2A−2A)+e2x (4A−A) = 3Ae2x . For this to be a solution, it must equal e2x , so we must have A = 1/3. The solution then is 1 y (x) = c1 e−x + c2 e2x + xe2x 3 2 ...
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This note was uploaded on 09/17/2011 for the course MATH 54 taught by Professor Chorin during the Spring '08 term at University of California, Berkeley.

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math54 - quiz 9 - solns - Math 54 quiz 9 Solutions 1. Find...

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