I'm a bit stuck on a homework problem laid out by my professor. It's a relatively standard 2nd order ODE, but she's asked us to use a change of variable to solve it, which is throwing me off quite a bit. The problem is as follows.
$$ xy'' + 2y' + xy =0, y(\pi ) = -1, y'(\pi ) = 2 $$
The change of variables provided is
$$ u=xy $$
Any help at all as to how to go about this would be greatly appreciated. Thanks!
$$ u=xy\implies u'=y+xy' \implies u''=2y' +xy'' $$
so your DE becomes ...
$$ u''+u=0; u(\pi)=-\pi, u'(\pi)=2\pi - 1 $$