If the eigen functions corresponding to the distinct eigen values of the Sturm-Liouville problem $$y''-3y'=\lambda y$$ are orthonormal with respect to weight function $w(x)$ then what is the value of $w(x)$ ?
(A) $e^{-3x}$
(B) $e^{-2x}$
(C) $e^{2x}$
(D) $e^{3x}$
Taking $\lambda =-k^2$, I got the solution of the given DE (when $4k^2\ge 9$) is
$$y(x)=e^{3x/2}\left\{A\cos \frac{x\sqrt{4k^2-9}}{2}+B\sin \frac{x\sqrt{4k^2-9}}{2}\right\}$$
Then $y(0)=0$ gives $A=0$ and then we get the eigen values are $\lambda =-\frac 14(4n^2+9)$ and the eigen functions are $\displaystyle e^{3x/2}\sin (nx)$.
But when I check for orthogonality, I can not find any $w(x)$ from the given options so that
$$\int_0^\pi w(x)e^{3x/2}\sin (nx) \,dx =0.$$
Where is my fault ? Any hint. or suggestion please ?
Best Answer
You have to make the equation self-adjoint. The obstacle is the first-derivative term. To absorb this into the second-derivative term, you need $e^{-3x}$ as integrating factor $$ (e^{-3x}y')'=λe^{-3x}y. $$ Generally you want to find the normal form $$ (p(x)y'(x))'+q(x)y(x)=λw(x)y(x) $$ of the Sturm-Liouville equation.
You could also see this from your solution formulas. Note that the weight function is for a scalar product, there are two eigenfunctions as factors in the integrand, $$ \int_0^\pi w(x)y_k(x)y_n(x)\,dx=\int_0^\pi w(x)e^{3x}\sin(kx)\sin(nx)\,dx $$