Find a particular solution to the non-homogeneous differential equation
$$y''+4y'+5y=−15x+e^{−x}$$
I got the homogeneous solution as :
$$
c_1e^{-2x}\cos(x)+c_2e^{-2x}\sin(x)
$$
I am having trouble finding the particular solution
[Math] Find a particular solution to the non-homogeneous differential equation $y”+4y’+5y=−15x+e^{−x}$
ordinary differential equations
Best Answer
Thinks of a function $p$ such that when you apply the operator $(D^2+4D+5I)$ to it, you end up with $-15x+e^{-x}$. Surely something like this would have to have the form $$p(x) = ae^{-x} + bx + c.$$ Then just plug it in to $y''+4y'+5y$, you'll get $$(D^2+4D+5I)p = 2ae^{-x}+5bx+4b+5c.$$ If you put $a=\frac12$, $b=-3$ and $c=\frac{12}5$, you'll end up with $-15x+e^{-x}$, thus the particular integral you should take is $$\boxed{p(x) = \tfrac12e^{-x}-3x+\tfrac{12}5.}$$
This method (and why it works) is explained on pages 2-6 of these notes.