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Quiz #2 Question (10 points total) (a) Use the reduction of order method to find a second solution of the differential equation y 00 + 16y = 0 given that y1 (x) = cos(4x) is a solution. (b) Given that yp1 = 3e2x and yp2 = x2 + 3x are, respectively, particular solutions of the differential equations y 00 − 6y 0 + 5y = −9e2x and y 00 − 6y 0 + 5y = 5x2 + 3x − 16 find a particular solution of the differential equation y 00 − 6y 0 + 5y = −10x2 − 6x + 32 + e2x . Solution: (a) Set y = u(x) cos(4x). So y 0 = −4u sin(4x) + u0 cos(4x) and y 00 = u00 cos(4x) − 8u0 sin(4x) − 16u cos(4x). Thus y 00 + 16y = cos(4x)u00 − 8 sin(4x)u0 = 0 or u00 − 8 tan(4x)u0 = 0. Now set w = u0 to get w0 − 8 tan(4x)w = 0. This is linear. An integrating factor is e−8 R tan(4x)dx = cos2 (4x). So d 2 cos (4x)w = 0 dx and so cos2 (4x)w = c. Thus u0 = c sec2 (4x) and so u = c1 tan(4x). So y = u(x)y1 (x) = tan(4x) cos(4x). (b) 1 1 yp = −2(x + 3x) + (3e2x ) = −2x2 − 6x − e2 x. 9 3 2