Question

y''+2y' + λy=0, y(0)=0, y(1)=0

Answer 1

I'll assume you don't know the Laplace transform yet.

Characteristic equation:
\[r^2+2r+\lambda=0\]
which has roots \(r_1\) and \(r_2\). Thus the general solution will have the form \(C_1e^{r_1 t}+C_2e^{r_2 t}\).

Answer 2

Hmm, hold up, are you sure about those initial conditions? Do you mean \(y'(0)=1\), or \(y'(1)=0\), or is this a boundary value problem?

Answer 3

I'd wager that it's the latter. In that case, you'll be using the given boundary values to determine \(C_1\) and \(C_2\) by plugging in \(t=0,~y=0\) and \(t=1,~y=0\).