Solving 2nd-order linear non-homogeneous differential equation:
y^2 - 38*y^1 + 361*y = 18*exp(19x) ; y^2 = double prime ; y^1 = single prime
finding the particular solution from yp(x) = a*x*exp(19x)
taking double prime and single prime and substituting them back into the original equation, gives you the equation below:
{(361ax+38a)*exp(19x)}+{(-722ax-38a )*exp(19x)}+{(361ax)*exp(19x)} = 18*exp(19x)
then, divide both side by exp(19x)
(361ax+38a) + (-722ax-38a) + (361ax) = 18
then, add and subtract like terms
0 = 18
can all the like terms cancel before the undetermined coefficients can
be found?