Complex number equivalency

Question

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.

Answer 1

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:

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