Find a particular solution to the non-homogeneous differential equation $y''+4y'+5y=−15x+e^{−x}$
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
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$\begingroup$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.
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