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203_pdfsam_math 54 differential equation solutions odd

# 203_pdfsam_math 54 differential equation solutions odd -...

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Exercises 4.5 By the method of undetermined coeﬃcients, a particular solution y p ( x ) to the original equation has the form y p ( x ) = x s ( A cos x + B sin x ) e x . We choose s = 0 because r = 1 + i is not a root of the auxiliary equation. So, y p ( x ) = ( A cos x + B sin x ) e x y p ( x ) = [( A + B ) cos x + ( B A ) sin x ] e x y p ( x ) = (2 B cos x 2 A sin x ) e x . Substituting these expressions into the equation, we compare the corresponding coeﬃcients and find A and B . { (2 B cos x 2 A sin x ) 3[( A + B ) cos x + ( B A ) sin x ] + 2( A cos x + B sin x ) } e x = e x sin x ( A + B ) cos x + ( A B ) sin x = sin x A + B = 0 , A B = 1 A = 1 / 2 , B = 1 / 2 . Therefore, y p ( x ) = (cos x sin x ) e x 2 and y ( x ) = (cos x sin x ) e x 2 + c 1 e x + c 2 e 2 x is a general solution to the given nonhomogeneous equation. 21. Since the roots of the auxiliary equation, which is
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