Boundary Value Problem: Solving with Eigenvalues and Eigenvectors

[Julio1]

Solve the boundary value problem:

$\left\{
\begin{array}{lcl}
y''&=&0,\hspace{1.0mm} 1<x<2\\
y(1)&=&0\\
y(3)+y'(3)&=&0
\end{array}
\right.
$

For the problem, I first calculate the eigenvalues and after check the roots and finally find the eigenvectors. Is correct this? Help me please :).

 

[evinda]

Solve the boundary value problem:

$\left\{
\begin{array}{lcl}
y''&=&0,\hspace{1.0mm} 1<x<2\\
y(1)&=&0\\
y(3)+y'(3)&=&0
\end{array}
\right.
$

For the problem, I first calculate the eigenvalues and after check the roots and finally find the eigenvectors. Is correct this? Help me please :).

(Wave)

In this case, we can just notice that $y''=0 \Rightarrow y(x)= c_1 x+ c_2, c_1, c_2 \in \mathbb{R}$.

And now it remains to use the given boundary conditions.

Is there at this point: $y(3)+y'(3)=0$ maybe a typo? Since it is given that $1<x<2$.

 

[Julio1]

Thanks evinda :).

The characteristic polynomial is $p(r)=r^2$, hence that $r=0$ is only root. Therefore, have that $y(x)=c_1e^{r_1 x}+c_2xe^{r_2 x}=c_1+c_2x$, for $c_1,c_2\in \mathbb{R}.$ Follow that $y'(3)=c_2$, so, $c_1+c_2=0.$ In conclusion, $c_1=c_2=0$, therefore, $y(x)=0$ is the trivial solution.