# Second order homogeneous ODE

## Homework Statement

Find the values of $α$ for which all the solutions of $y''-(2α-1)y'+α(α-1)y=0$ (a) tend to zero and (b) are ilimited, when $t->∞$.

## Homework Equations

$y''-(2α-1)y'+α(α-1)y=0 => (t)=Ae^{αt}+Be^{(α-1)t}$

## The Attempt at a Solution

I found that the general solution to the problem is $y(t)=Ae^{αt}+Be^{(α-1)t}$, which I believe is correct. Then I said that (a) is verified for $t<1/2$ and (b) for $t>=1/2$, but the book's answer is (a) $t<0$ and (b) $t>1$.

What am I missing?
Thanks

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Simon Bridge
Homework Helper
As you wrote it, the question asks for values of $\alpha$, not $t$.
So it looks like your (and the book's) answer is for a different question.

$y(t)\rightarrow 0$ for $t\rightarrow \infty$ when $\alpha < 0$ which makes y(t) a sum of decaying exponentials.

My mistake, I meant α where I said t. I understand what you said, but then what happens for 0<α<1 ?

Thanks

Simon Bridge
Well then put that into y(t) ... you have two exponentials added together and $\alpha$ appears in the power.