Find the non-trivial eigen functions

Question

I am asked to find the non-trivial eigen values for the following differential equation

$y''+2y'+\lambda y=0$

given $$y'(0)-y(0)=0,y'(1)=0$$

So the characteristic equation is

$r^2+2r'+\lambda =0$

solving, I get $r_{1,2}=-1 \pm\sqrt{1-\lambda}$

Then if I go ahead and do my cases, I get that when ${1-\lambda}=0$ there is a trivial solution.

If I do the other cases, I don't get the boundary conditions enabling me to solve for one of the constants in any case. Normally, one of the conditions would allow me to immediately solve for one of the constants but in this case, I can't figure out what I'm doing wrong.

For ${1-\lambda}=0$, I have

$y=c_{1}e^{-t}+c_{2}te^{-t}$

For ${1-\lambda} > 0$, I have

$y=c_{1}e^{(-1 +\sqrt{1-\lambda})t}+c_{2}e^{(-1 -\sqrt{1-\lambda})t}$

and for For ${1-\lambda} < 0$, I have

$y=c_{1}e^{-t}\cos(\sqrt{1-\lambda}t)+c_{2}e^{-t}\sin(\sqrt{1-\lambda}t)$

Any help is very much appreciated.

Answer 1

Set $λ=1-r^2$ to get the solution as $y(t)=c_1e^{(-1-r)t}+c_2e^{(-1+r)t}$ and then \begin{align} y'(t)&=(-1-r)c_1e^{(-1-r)t}+(-1+r)c_2e^{(-1+r)t}\\ 0=y'(0)-y(0)&=(-2-r)c_1+(-2+r)c_2\\ 0=y'(1)&=(-1-r)c_1e^{-1-r}+(-1+r)c_2e^{-1+r} \end{align} Isolating $-c_2/c_1$ gives the equation $$ \frac{2+r}{2-r}=e^{-2r}\frac{1+r}{1-r} $$ which has a solution close to $0.871180357493748$ assuming $r$ is real.

For imaginary $r=iω$ write the equation as $$ (2-iω+ω^2)(\cosω+i\sinω)=(2+iω+ω^2)(\cosω-i\sinω) \\~\\\iff\\~\\ (2+ω^2)\sinω-ω\cosω=0 \\~\\\implies\\~\\ \sin\left(ω-\frac{ω}{2+ω^2}\right)=O(\frac1{ω^2}) $$ which has roots close to the roots of the sine, i.e., close to $ω_0=k\pi$, $k=1,2,3,...$. Using the next approximation order gives root approximations $ω_1=k\pi(1+\frac1{1+(k\pi)^2})$.

k         omega1            final root approx.
-------------------------------------------------
1    3.43061813581203        3.3878600880634 
2    6.43840840331423        6.4304471378259
3    9.52970005276935        9.527119698449
4   12.6454473267738        12.6443221099437
...
10  31.4477253056264        31.4476505886781