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

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.

Yet prevent ...

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