Series Solution to $y''+xy=e^x$ I am thoroughly familiar with using power series to solve the differential equation $y''+xy=0$, but how exactly does one go about solving $y''+xy=e^x$?
I would imagine you represent $e^x$ as it's power series, along with everything else first, but then what?

Just looking at
$y''+xy=e^x$
and ignoring the part
about power series,
I would first let
$y = ze^x$.
Then
$y'
=e^x(z+z')
$
and
$y''
=e^x((z+z')+(z'+z''))
=e^x(z+2z'+z'')
$
so the equation becomes,
dropping the $e^x$,
$1
=z''+2z'+z+xz
=z''+2z'+z(1+x)
$.
From this,
we can readily get
a recurrence for
the coefficients of
$z$.
