Need help solving a Non-Homogenous DE

[juice34]

I need help solving a higher order differential equation by reduction of order.
It will be greatly appreciated if all steps are posted as well!

y(Double Prime)-3y(Prime)+2y=5e^3x where y sub one =e^x.

 

[lurflurf]

"It will be greatly appreciated if all steps are posted as well! "
um no, well maybe with the best bits obfuscated :)
let D be differentiation w/ respect to x
let y=u(x)*exp(x)
L=(D^2-2D+2)
Ly=5exp(3x)
or
(D^2-2D+2)y=5exp(3x)
provided Lexp(x)=0 (is it? you should know)
we will have transformed the equation (by substitution) into
(aD^2+bD+c)u=f(x)
where c=0 (show this is true in general)
then
u=[(aD^2+bD)^-1] f(x)
or if you prefer find an integrating factor v(x) so that
v(x)(aD^2+bD)u(x)=D(v(x)Du(x))
thus
(aD^2+bD)u(x)=f(x)
becomes
D(v(x)Du(x))=v(x)*f(x)
and easy to solve ie
u(x)=(1/D)(1/v(x))(1/D)[v(x)*f(x)]
notice how each (1/D) yeilds one arbitrary constant as expected