Ordinary Differential Equations

[Combinatorics]

Homework Statement

Find all eigenvalues and eigenfunctions:
-y'' (x) = \lambda y(x) , x \in (a,b)
y(a)= y(b) =0.

Homework Equations

sin x = \frac{e^{ix} + e^{-ix} }{2i}.

The Attempt at a Solution

So actually the only problem I have is to find the eigenfunctions (which should be something like sin \{ \frac{n \pi (x-a) }{(b-a)} \} ) .
I received the eigenvalues are: \lambda_n = \frac{n^2 \pi^2 }{(b-a)^2}.
But how can I solve the two equations system I receive when applying these eigenvalues to the solution: y= c_1 e^{i \sqrt{\lambda_n} x} + c_2 e^{-i \sqrt{\lambda_n} x}.

I hope someone will be able to help me solve this two equations-system.

Thanks in advance!

 

[LCKurtz]

Homework Statement

Find all eigenvalues and eigenfunctions:
-y'' (x) = \lambda y(x) , x \in (a,b)
y(a)= y(b) =0.

Homework Equations

sin x = \frac{e^{ix} + e^{-ix} }{2i}.

The Attempt at a Solution

So actually the only problem I have is to find the eigenfunctions (which should be something like sin \{ \frac{n \pi (x-a) }{(b-a)} \} ) .
I received the eigenvalues are: \lambda_n = \frac{n^2 \pi^2 }{(b-a)^2}.
But how can I solve the two equations system I receive when applying these eigenvalues to the solution: y= c_1 e^{i \sqrt{\lambda_n} x} + c_2 e^{-i \sqrt{\lambda_n} x}.

I hope someone will be able to help me solve this two equations-system.

Thanks in advance!

Write the problem as $y'' +\lambda y = 0$ and consider the case where $\lambda = \mu^2 > 0$ so $y''+\mu^2y=0$. This gives you the solution pair $\{e^{i\mu x},e^{-i\mu x}\}$. But by using appropriate linear combinations of those you could use the linearly independent pair $\{\sin(\mu x),\cos(\mu x)\}$ instead. Even better, you can use any appropriate linear combination of those to get something easier to work with. Try using this pair: $\{\cos(\mu(x-a)),\sin(\mu(x-a))\}$. Take a general linear combination of those two and apply your boundary conditions. It will all drop out. At the end, remember $\lambda = \mu^2.$

 

[Combinatorics]

Is there any way of solving the two equations without having to do the process you mentioned?

Thanks !

 

[HallsofIvy]

One can, by direct substitution, show that y(x)= Acos(\lambda (x- a)+ Bsin(\lambda (x- a)) is the general solution to the differential equation y''+ \lambda^2 y= 0, for any number a. Of course, then y(0)= A= 0.
Now y(b)= B sin(\lambda(b- a))= 0. You know, I presume that sin(x) is 0 if and only if x is a multiple of \pi so you use that to determine possible values for the eigenvalue, \lambda.

 

[Combinatorics]

Great ! Thanks ! But how can I prove that this is the general solution ? (i.e.- it obviously solves our problem...but why is it the general solution? why is it the only form of our eigenfunctions? )

Thanks!

 

[LCKurtz]

Is there any way of solving the two equations without having to do the process you mentioned?

Thanks !

I suppose so, but I don't know why you would want to. Certainly you would use $y = A\cos(\mu x) + B\sin(\mu x)$ instead of the exponential version. I will show you how to get the solution from there, assuming you didn't think of the $(x-a)$ version. Applying the boundary conditions you get:$$A\cos(\mu a) + B\sin(\mu a)=0$$ $$
A\cos(\mu b) + B\sin(\mu b)=0$$For this to have a non-trivial solution for $A$ and $B$ the determinant of coefficients must be 0:$$
\left |\begin{array}{cc}
\cos(\mu a) & \sin(\mu a)\\
\cos(\mu b) & \sin(\mu b)
\end{array}\right | =\cos(\mu a)\sin(\mu b)-\cos(\mu b)\sin(\mu a)= 0$$ $$
\sin(\mu(b-a))=0$$
This gives $\mu(b-a) = n\pi$ or $\mu_n = \frac{n\pi}{b-a}$

Under this condition the system of equations is dependent, so just using the first one we can write $A\cos(\mu a) = -B\sin(\mu a)$ so $A=-\frac{B\sin(\mu a)}{\cos(\mu a)}$. Substituting that for the $A$ in the general solution for $y$ gives $$
y_n = -B\frac{\sin(\mu_n a)}{\cos(\mu_n a)}\cos(\mu_n x) + B\sin(\mu_n x)
=(\frac{B}{\cos(\mu_n a)})(-\sin(\mu_n a)\cos(\mu_n x)+\sin(\mu_n x)\cos(\mu_n a))
=(\frac{B}{\cos(\mu_n a)})\sin(\mu_n(x-a))$$Since any constant times a solution is a solution, we can ignore the constant $\frac{B}{\cos(\mu_n a)}$ and take$$
y_n = \sin(\mu_n(x-a))$$So what I showed you in my first post is something worth learning. It saves a lot of work.

 

[Combinatorics]

Thanks a lot !