Can someone help me with this?
How do I find the eigenvalues and eigenfunctions for the Sturm-Liouville problem
$\displaystyle x^2y''+xy'+3y=uy$;
$\displaystyle y(1)=0, y(2)=0$
Thanks in advance.
That equation is the same as $\displaystyle x^2y"+ xy'+ (3-u)y= 0$, an "Euler type" or "equipotential" equation. The change of variable t= ln(x) changes it to an equation with constant coefficients. You should be able to find the general solution. Obviously, y(x)= 0, for all x, satifies the equation as well as y(1)= y(2)= 0. The eigenvalues are the values of u for which there are other, non-trivial, solutions. And the eigenfunctions are the non-trivial solutions.
The problem with that is that in order that the boundary conditions to be satisfied your m must be complex! I think most people, who have studied d.e.s with constant coefficients will be more comfortable with $\displaystyle e^{(a+bi)t}= e^{at}(cos(bt)+ i sin(bt))$ than with $\displaystyle x^{a+bi}$.
In my experience, I find students have more of a problem transforming the DE under $\displaystyle x = \ln t$, than using $\displaystyle y = x^m$. Once the basic solutions are derived $\displaystyle x^a \sin b \ln x$ and $\displaystyle x^a \cos b \ln x$, I find the students find these easier to use.
Hey I'm trying to answer the exact same problem for homework and am really confused..I get as far as;
$\displaystyle y = Ax^{ \sqrt{(3 - u)}} + Bx^{-\sqrt{(3 - u)}} $
But then I'm not sure what to do - I can see I'm going to get different results if u >3, <3, =3, but I don't know how to put that into an overall answer... at what point should I be using the boundary conditions?
Also, the 2nd part of the question asks me to expand a hypothetical f(x) in terms of the eigenfunctions - I assume in the format $\displaystyle f(x) = \Sigma{(Ck)Yk} $ where Ck are the sturm-louiville coefficients. But how can I do this if the format of Yk changes depending on whether u is > or < 3? Confused... :-s
The best thing to do is consider the following three cases:
$\displaystyle u = 3 + \omega^2,\;\; u = 3,\;\; u = 3 - \omega^2$. That corresponds to your three cases.
Cases 1: $\displaystyle u = 3 + \omega^2$ then
$\displaystyle y = A x^{\omega} + B x^{-\omega}$
Now use your BC's.
$\displaystyle y(1) = A 1^{\omega} + B 1^{-\omega} = 0$
$\displaystyle y(2) = A 2^{\omega} + B 2^{-\omega} = 0$
Two equations for the two unknowns $\displaystyle A\;\; \text{and}\;\;B$. The first says $\displaystyle B = -A$ whereas the second
$\displaystyle A 2^{\omega} -A 2^{-\omega} = 0$ so $\displaystyle A \left(2^{\omega} - 2^{-\omega}\right) = 0$ which gives A=0 leading to the trival solution $\displaystyle y = 0$. Similarly for the second case. It's the third case that's interesting. You try now.