Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The equation $x^{2}y''+xy'+5y=0$ is equidimensional and has the general solution $y(x) = c_{1}\cos(\sqrt{5}\log x) + c_{2}\sin(\sqrt{5}\log x)$. But this differential equation also has a regular singular point around $x = 0$ and hence we can use the Frobenius method to find $a_{n}$ such that $y(x) = \sum_{n = 0}^{\infty} a_{n}x^{n + s}$. But wouldn't this imply that $\sin(\sqrt{5}\log x)$ has a series expansion around $x = 0$? Also when I use the indicial equation to find $s$, I get $s = \pm i\sqrt{5}$ which doesn't seem right?

share|improve this question

1 Answer 1

Remember $\sin(u)=(e^{iu}-e^{-iu})/(2i)$, so of course substituting $u=\sqrt{5}\log x$ you'll get that the function is a difference of two powers of $x$, namely the powers $s=\pm i\sqrt{5}$ as you solved for.

share|improve this answer
    
Not quite relevant to the question but I want to say that anon's stand alone complex avatar is awesome. –  Shuhao Cao Dec 28 '11 at 21:19
    
RIAA would have a different viewpoint on that matter ;-) He would be better off by scrambling it into the bits of some cat photo ;-) –  SasQ Oct 30 '13 at 16:15

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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