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 .W echoo se 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 0 p ( x )=[( A + B )cos x +( B A )sin x ] e x y 0 p ( x )=(2 B cos x 2 A sin x ) e x . Substituting these expressions into the equation, we compare the corresponding coefficients and Fnd 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 =s in 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
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