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!


1 Answer 1


$$ u=xy\implies u'=y+xy' \implies u''=2y' +xy'' $$

so your DE becomes ...

$$ u''+u=0; u(\pi)=-\pi, u'(\pi)=2\pi - 1 $$

  • $\begingroup$ Where do you 'substitute back'? I've obtained a general form for u, but I'm getting stuck on how to transfer that back to an equation involving x and y. $\endgroup$
    – user387187
    Nov 8, 2016 at 5:29
  • $\begingroup$ $y(x)$ is simply $u(x)/x$. $\endgroup$
    – Chee Han
    Nov 8, 2016 at 5:57

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