Hi I am having some problems with the simplification of the DE, I am fine up on till $$y_p(x)=v_1(x)y _ p1 (x) + v_2(x)y _ (x) $$

$$ \ frac12 e ^ \ left(\ frac 4-\ frac x2 \ right)+ \ frac 12e ^ \ left(\ frac 4+\ frac x2 \ right)$$I can not streamline it to obtain $\ frac x2 \ sinh(x)$ I would certainly value some information preferably


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The simplification is challenging if you aren"t searching for it. When you integrate the corresponding remedy with the specific service, you obtain rid of the $\ tfrac14 \ cosh(x)$.

Simplification where you are:

Team like terms:: $$\ tfrac14x(e ^ x - e ^ -x) - \ tfrac18(e ^ x + e ^ )$$This streamlines to:: $$\ tfrac12x \ sinh(x) - \ tfrac18(e ^ x + e ^ -x )$$You wear"t intend to streamline the last term right into $-\ tfrac14 \ cosh(x)$ due to the following:$$Y = Y_c + Y_p. \ \ Y = (A)e ^ + (B)e ^ x - \ tfrac18(e ^ x + e ^ -x) + \ tfrac12x \ sinh(x). \ \(A - \ tfrac18)e ^ -x = Ce ^ \ \ (B - \ tfrac18)e ^ x = De ^ .$$The approximate consistent from the free of charge service integrated with the $-\ tfrac18$ from the certain service can be streamlined as one more approximate constant.
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2 various outcomes on addressing a differential formula by variant of criteria V.S unclear coefficients
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