Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm a bit confused over the solution to a complex ode: $i\alpha y = \beta y''$

The solution to the characteristic polynomial is $r = \pm \sqrt{i\alpha/\beta}$. Somehow my book is getting the solution: $r = \pm (i+1)\sqrt{\alpha/2\beta}$.

Unfortunately, I haven't had a course in complex numbers, so they are a bit of a weak spot for me. Can anyone explain why the two forms are equivalent and how to get from the former to the latter?

This isn't a homework problem or anything, just one step of many in a derivation and I'm trying to fully understand each part.

share|cite|improve this question
up vote 5 down vote accepted

Note that $(i+1)^{2} = i^{2} + 2i + 1 = -1 + 1 + 2i = 2i$, so in fact, the two roots you've listed are equal.

Just to make it perfectly explicit:

$$r = \pm(i+1)\sqrt{\frac{\alpha}{2\beta}} = \pm\sqrt{\frac{(i+1)^{2}\alpha}{2\beta}} = \pm\sqrt{\frac{2i\alpha}{2\beta}} = \pm\sqrt{\frac{i\alpha}{\beta}}$$

share|cite|improve this answer

Your Answer


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.