Consider the DE: $y''$ + $2\lambda y'$ + $\lambda^2$$y$ = $0$ subject to boundary conditions: $y(1) + y'(1) = 0$ and $3y(2) + 2y'(2) = 0$. The problems asks to find eigenvalues and eigenfunctions of the given BVP.
My approach: Obviously, The characteristic eqn gives $r = - \lambda$. So general solution is $y(x) = c_1e^{-\lambda x} + c_2xe^{-\lambda x}$. and by applying boundary condition, I obtain the following system:
$c_1(1 - \lambda) + c_2(2 - \lambda) = 0$
$c_1(3 - 2\lambda) + 4c_2(1 - \lambda) = 0$
From here, Im having difficulties in obtaining eigenvalues. I would to ask If my approach is correct or If I am probably doing something incorrect. Is there a better way to solve this problem?
thanks