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Amcote
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Homework Statement
Consider the following Sturm-Liouville Problem:
[tex]\dfrac{d^2y(x)}{dx^2} + {\lambda}y(x)=0, \ (a{\geq}x{\leq}b)[/tex]
with boundary conditions
[tex]a_1y(a)+a_2y{\prime}(a)=0, \ b_1y(b)+b_2y{\prime}(b)=0[/tex]
and distinguish three cases:
[tex]a_1=b_1, a_2{\neq}0, b_2{\neq}0[/tex][tex]a_2=b_2=0, a_1{\neq}0, b_1{\neq}0[/tex][tex]a_21{\neq}0, a_2{\neq}0, b_1{\neq}0, b_2{\neq}0[/tex]
a) Without determining the actual eigenvalues and eigenfunctions, prove, in detail, the orthogonality of non-degenerate (i.e. having different eigenvalues) eigenfunctions in all three cases.
Homework Equations
Suppose we have a solution with eigen functions [tex]y_j,y_m[/tex] such that their eigen values are [tex]\lambda_j,\lambda_m, \lambda_m\neq{\lambda}_j[/tex]
Then they are orthogonal if:
[tex]\int_{a}^{b}dxy_j(x)y_m(x)=0, [/tex]
The Attempt at a Solution
So I am okay with doing this integral and proving the orthogonality of the two eigen functions. But what i do not understand is what a1, a2, b1 and b2 are and how I implement the special cases.
Basically I solve the integral and I am left with
[tex][y_{j}p(x)y_m{\prime}(x)-y_m(x)p(x)y_j{\prime}(x)]^{b}_{a}-\int^{b}_{a}[y_{j}{\prime}(x)p(x)y_{m}{\prime}(x)-y_{m}{\prime}(x)p(x)y_{j}{\prime}(x)][/tex]
where clearly the second part is zero and the first part should be zero with boundary conditions. But like I said I have no idea what the a1, a2, b1, b2 mean.
Any help would be appreciated