Worksheet11_1B_Solutions

# Worksheet11_1B_Solutions - Math 1B Worksheet 11 Solutions...

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Math 1B Worksheet 11 Solutions Variation of Parameters Shortcut Let’s say we want to solve ay 00 + by 0 + cy = G ( x ) using variation of parameters. Suppose that the general solution to ay 00 + by 0 + cy = 0 is y = c 1 y 1 + c 2 y 2 (for example, if our equation was y 00 + 2 y 0 + y = 0, then we would have y 1 = e - x and y 2 = xe - x ). We wish to ﬁnd a particular solution of the form u 1 y 1 + u 2 y 2 . This can be accomplished by solving the following system of equations: au 0 1 y 0 1 + au 0 2 y 0 2 = G ( x ) u 0 2 y 1 + u 0 2 y 2 = 0 In the aforementioned example, this system would look like: - u 0 1 e - x + u 0 2 ( e - x - xe - x ) = G ( x ) u 0 2 e - x + u 0 2 xe - x = 0 1. For each of the following possibilities for G ( x ), decide whether you would use undetermined coeﬃcients or variation of parameters in solving the diﬀerential equation ay 00 + by 0 + cy = G ( x ). For those where you choose undetermined coeﬃcients, write the form of your guess: a) e - 3 x . Undetermined coeﬃcients; y p = Ae - 3 x . b) e x 2 . Variation of parameters. c) sin 3 x . Undetermined coeﬃcients; y p = A cos 3 x + B sin 3 x . d) cos x - sin 5 x . Undetermined coeﬃcients; y p = A cos x + B sin x + C cos 5 x + D sin 5 x . e) sin x/x . Variation of parameters. f) x 3 e - 7 x sin 6 x + 7 - x 3 cos 4 x . Undetermined coeﬃcients; y p = e - 7 x ( Ax 3 + Bx 2 + Cx + D ) cos 6 x + e - 7 x ( Ex 3 + Fx 2 + Gx + H ) sin 6 x + I + ( Jx 3 + Kx 2 + Lx + M ) cos 4 x + ( Nx 3 + Px 2 + Rx + Q ) sin 4 x . 1

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g) e x / ( x 2 + 1). Variation of parameters. h) x + 3 + e x . Undetermined coeﬃcients; y p = Ax + B + Ce x .
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## This note was uploaded on 01/02/2011 for the course MATH 1B taught by Professor Reshetiken during the Spring '08 term at Berkeley.

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Worksheet11_1B_Solutions - Math 1B Worksheet 11 Solutions...

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