why Mathematica gives zero as eigenvalue for $y''+\lambda y=0$ for these boundary conditions?

Question

I was trying to verify my hand solution using Mathematica for this problem

$$ y^{\prime\prime}+\lambda y=0 $$

The domain is $0<x<L$ and the boundary conditions are

\begin{align*} y(0)+y^{\prime}(0)&=0\\ y(L)+y^{\prime}(L)&=0 \end{align*}

My hand solution gives these eigenvalues

$$ \lambda=\left\{ -1,\frac{\pi^{2}}{L^{2}},\frac{4\pi^{2}}{L^{2}},\frac {9\pi^{2}}{L^{2}},\frac{16\pi^{2}}{L^{2}},\cdots\right\} $$

Mathematica agrees with the above, but it also claims zero is an eigenvalue

$$ \lambda=\left\{ -1,0,\frac{\pi^{2}}{L^{2}},\frac{4\pi^{2}}{L^{2}},\frac {9\pi^{2}}{L^{2}},\frac{16\pi^{2}}{L^{2}},\cdots\right\} $$

Which I think is wrong. zero should not be an eigenvalue.

The question is: Is Mathematica result wrong? Am I reading Mathematica output wrong?

Mathematica output

ClearAll[L0, x, y, lam];
bc = {y'[0] + y[0] == 0, y'[L0] + y[L0] == 0};
Assuming[L0 > 0 && Element[lam, Reals], 
 DSolve[{y''[x] + lam y[x] == 0, bc}, y[x], x]]

The problem that I see is $n\geq 0$ in the above. Using the above result:

 tab1 = Table[(4*n^2*Pi^2)/L0^2, {n, 0, 4}];    
 tab2 = Table[(Pi^2 + 4*n*Pi^2 + 4*n^2*Pi^2)/L0^2, {n, 0, 4}]; 
 Union[tab1,tab2]

You see zero eigenvalue shows up, from applying it to (4 n^2 Pi^2)/L0^2. Mathematica result will match my hand solution if it said $n>0$ for (4 n^2 Pi^2)/L0^2 and said $n\geq 0$ for the second part (Pi^2 + 4 n Pi^2 + 4 n^2 Pi^2)/L0^2:

tab1 = Table[(4*n^2*Pi^2)/L0^2, {n, 1, 4}]; (*fixed. make it start from 1*)
tab2 = Table[(Pi^2 + 4*n*Pi^2 + 4*n^2*Pi^2)/L0^2, {n, 0, 4}]; 
Union[tab1,tab2]

Version 11.2 on windows 7.

Update

TO answer comment below, Here is the hand solution. The question was just misunderstanding on my part reading result of DSolve, since $\lambda=0$ can not be an eigenvalue, but it also showed there in the solution to DSolve, so I was asking why.

But DSolve is allowed to return trivial solution $y(x)=0$ which is not allowed when finding the eigenvalues. Lesson of the day: Use DEigenvalues to find eigenvalues instead of DSolve.

NDEigenvalues[{-y''[x]+NeumannValue[-y[x],x==1]+NeumannValue[y[x],x==0]},y[x],{x,0,1},6]

L0=1;
tab1=Table[ n^2 Pi^2/L0,{n,{1,2,3,4,5}}]//N;
Join[{-1},tab1]

Hand solution

Assume the solution is $y=Ae^{rx}$, then the characteristic equation is

\begin{align*} r^{2}+\lambda & =0\\ r & =\pm\sqrt{-\lambda} \end{align*}

Assuming $\lambda<0$

In this case $-\lambda$ is positive and hence $\sqrt{-\lambda}$ is also positive. Let $\sqrt{-\lambda}=\mu$ where $\mu>0$. Hence the roots are $\pm \mu$. This gives the solution

$$ y=c_{1}\cosh\left( \mu x\right) +c_{2}\sinh\left( \mu x\right) $$

Hence

$$ y^{\prime}=\mu c_{1}\sinh\left( \mu x\right) +\mu c_{2}\cosh\left( \mu x\right) $$

Left B.C. gives

\begin{equation} 0=c_{1}+\mu c_{2}\tag{1} \end{equation}

Right B.C. gives

\begin{align*} 0 & =c_{1}\cosh\left( \mu L\right) +c_{2}\sinh\left( \mu L\right) +\mu c_{1}\sinh\left( \mu L\right) +\mu c_{2}\cosh\left( \mu L\right) \\ & =\cosh\left( \mu L\right) \left( c_{1}+\mu c_{2}\right) +\sinh\left( \mu L\right) \left( c_{2}+\mu c_{1}\right) \end{align*}

Using (1) in the above, it simplifies to

$$ 0=\sinh\left( \mu L\right) \left( c_{2}+\mu c_{1}\right) $$

But from (1) again, we see that $c_{1}=-\mu c_{2}$ and the above becomes

\begin{align*} 0 & =\sinh\left( \mu L\right) \left( c_{2}-\mu\left( \mu c_{2}\right) \right) \\ & =\sinh\left( \mu L\right) \left( c_{2}-\mu^{2}c_{2}\right) \\ & =c_{2}\sinh\left( \mu L\right) \left( 1-\mu^{2}\right) \end{align*}

But $\sinh\left( \mu^{2}L\right) \neq0$ since $\mu^{2}L\neq0$ and so either $c_{2}=0$ or $\left( 1-\mu^{2}\right) =0$. $c_{2}=0$ results in trivial solution, therefore $\left( 1-\mu^{2}\right) =0$ or $\mu^{2}=1$ but $\mu ^{2}=-\lambda$, hence $\lambda=-1$ is the eigenvalue.

Corresponding eigenfunction is

$$ y=c_{1}\cosh\left( x\right) +c_{2}\sinh\left( x\right) $$

Using (1) the above simplifies to

\begin{align*} y & =-\mu c_{2}\cosh\left( x\right) +c_{2}\sinh\left( x\right) \\ & =c_{2}\left( -\mu\cosh\left( x\right) +\sinh\left( x\right) \right) \end{align*}

But $\mu=\sqrt{-\lambda}=1$, hence the eigenfunction is

$$ \fbox{$y\left( x\right) =c_2\left( -\cosh\left( x\right) +\sinh\left( x\right) \right) $} $$

Assuming $\lambda=0$

Solution now is

$$ y=c_{1}x+c_{2} $$

Therefore

$$ y^{\prime}=c_{1} $$

Left B.C. $0=y\left( 0\right) +y^{\prime}\left( 0\right) $ gives

\begin{equation} 0=c_{2}+c_{1} \tag{2} \end{equation}

Right B.C. $0=y\left( L\right) +y^{\prime}\left( L\right) $ gives

\begin{align*} 0 & =\left( c_{1}L+c_{2}\right) +c_{1}\\ 0 & =c_{1}\left( 1+L\right) +c_{2} \end{align*}

But from (2) $c_{1}=-c_{2}$ and the above becomes

\begin{align*} 0 & =-c_{2}\left( 1+L\right) +c_{2}\\ 0 & =-c_{2}L \end{align*}

Which means $c_{2}=0$ and therefore the trivial solution. Therefore $\lambda=0$ is not an eigenvalue.

Assuming $\lambda>0$

Solution is \begin{equation} y=c_{1}\cos\left( \sqrt{\lambda}x\right) +c_{2}\sin\left( \sqrt{\lambda }x\right) \tag{A} \end{equation}

Hence

$$ y^{\prime}=-\sqrt{\lambda}c_{1}\sin\left( \sqrt{\lambda}x\right) +\sqrt{\lambda}c_{2}\cos\left( \sqrt{\lambda}x\right) $$

Left B.C. gives

\begin{equation} 0=c_{1}+\sqrt{\lambda}c_{2} \tag{3} \end{equation}

Right B.C. gives

\begin{align*} 0 & =c_{1}\cos\left( \sqrt{\lambda}L\right) +c_{2}\sin\left( \sqrt{\lambda}L\right) -\sqrt{\lambda}c_{1}\sin\left( \sqrt{\lambda }L\right) +\sqrt{\lambda}c_{2}\cos\left( \sqrt{\lambda}L\right) \\ & =\cos\left( \sqrt{\lambda}L\right) \left( c_{1}+\sqrt{\lambda} c_{2}\right) +\sin\left( \sqrt{\lambda}L\right) \left( c_{2}-\sqrt {\lambda}c_{1}\right) \end{align*}

Using (3) in the above, it simplifies to

$$ 0=\sin\left( \sqrt{\lambda}L\right) \left( c_{2}-\sqrt{\lambda} c_{1}\right) $$

But from (3), we see that $c_{1}=-\sqrt{\lambda}c_{2}$. Therefore the above becomes \begin{align*} 0 & =\sin\left( \sqrt{\lambda}L\right) \left( c_{2}-\sqrt{\lambda}\left( -\sqrt{\lambda}c_{2}\right) \right) \\ & =\sin\left( \sqrt{\lambda}L\right) \left( c_{2}+\lambda c_{2}\right) \\ & =c_{2}\sin\left( \sqrt{\lambda}L\right) \left( 1+\lambda\right) \end{align*}

Only choice for non trivial solution is either $\left( 1+\lambda\right) =0$ or $\sin\left( \sqrt{\lambda}L\right) =0$. But $\left( 1+\lambda\right) =0$ implies $\lambda=-1$ but we said that $\lambda>0$. Hence other choice is \begin{align*} \sin\left( \sqrt{\lambda}L\right) & =0\\ \sqrt{\lambda}L & =n\pi\qquad n=1,2,3,\cdots\\ \lambda_{n} & =\left( \frac{n\pi}{L}\right) ^{2}\qquad n=1,2,3,\cdots \end{align*}

The above are the eigenvalues. The corresponding eigenfunction is from (A)

$$ y=c_{1_{n}}\cos\left( \sqrt{\lambda_{n}}x\right) +c_{2_{n}}\sin\left( \sqrt{\lambda_{n}}x\right) $$

But $c_{1_{n}}=-\sqrt{\lambda_{n}}c_{2_{n}}$ and the above becomes

\begin{align*} y\left( x\right) & =-\sqrt{\lambda_{n}}c_{2_{n}}\cos\left( \sqrt {\lambda_{n}}x\right) +c_{2}\sin\left( \sqrt{\lambda_{n}}x\right) \\ & =C_{n}\left( -\sqrt{\lambda_{n}}\cos\left( \sqrt{\lambda_{n}}x\right) +\sin\left( \sqrt{\lambda_{n}}x\right) \right) \end{align*}

Summary

Eigenvalue $\lambda=-1$ with eigenfunction $y\left( x\right) =c_{2}\left( -\cosh\left( x\right) +\sinh\left( x\right) \right) $ and eigenvalues $\lambda_{n}=\left( \frac{n\pi}{L}\right) ^{2},n=1,2,3,\cdots$ with eigenfunctions $C_{n}\left( -\sqrt{\lambda_{n}}\cos\left( \sqrt{\lambda_{n} }x\right) +\sin\left( \sqrt{\lambda_{n}}x\right) \right) $. Listing the eigenvalues in order:

$$ \lambda=\left\{ -1,\frac{\pi^{2}}{L^{2}},\frac{4\pi^{2}}{L^{2}},\frac {9\pi^{2}}{L^{2}},\frac{16\pi^{2}}{L^{2}},\cdots\right\} $$

Answer 1

The value lambda = 0 simply corresponds to the most trivial solution, but Mathematica never claims it to be an eigenvalue: it only claims that it's a possible solution to the differential equation you put into DSolve. There's no real reason why DSolve shouldn't give you the trivial solution.