4
$\begingroup$

Consider the equation $$(x+3)y''+2y'-4(x+3)y=0.$$ I was trying to solve it by finding the solution in the form of power series. However, I stuck while trying to find any regularity in the coefficients, while Wolfram Alpha provides a pretty much nice answer. Could you please point out the direction?

$\endgroup$
1
$\begingroup$

There is a life hack just for you: solve the thing with Wolfram, see how it could be simplified, then solve it the way you were supposed to. What if we switch to $u(x)=(x+3)\cdot y(x)$ and look for that in the form of power series?

$\endgroup$
  • $\begingroup$ This is a really helpful observation. I must admit I could not spot such a trick! $\endgroup$ – kraken kraken Sep 30 '15 at 22:41
0
$\begingroup$

You could certainly use power series, but I use the expansion about the point $x=-3$ rather than $x=0$ (since you have those $(x+3)$ terms). Alternatively, let $z(x)=(x+3)y(x)$, so $z'=(x+3)y'+y$ and $z''=(x+3)y''+2y'$. You can then find a fairly simple ODE for $z$ and solve it however you like.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.