Answer to Question #128660 in Differential Equations for Duaa

Question

put the following equation into sturm-liouville form x^2y^''+xy^'+(λx^2-n^2)y=0

Answer 1

$$Solution$$

Bessel’s equation

$x^2 y^{''}+xy^{'}+(\lambda x^2-n^2)y=0$

Theorem. Any second order linear operator can be put into the form of the Sturm-Liouville operator

This is in the correct form. We just identify $p(x) =a_2(x)$ and $q(x) =a_0(x)$ .However, considering the Bessel's equation;

$a_2(x) =x^2$ and $a^′_2(x) = 2x \ne a_1(x).$

In the Sturm Liouville operator the derivative terms are gathered together into one perfect derivative

We need only multiply this equation by

$$\frac{1}{x^2}\epsilon^{\int \frac{dx}{x}}=\frac{1}{x}$$

to put the equation in Sturm-Liouville form;

$\frac{x^2 y^{''}}{x}+\frac{xy^{'}}{x}+(\frac{\lambda x^2}{x}-\frac{n^2}{x})y=x y^{''}+y^{'}+(\lambda x-\frac{n^2}{x} )y=(xy^{'})^{'}+(\lambda x-\frac{n^2}{x})y=0$

$$(xy^{'})^{'}+(\lambda x-n^2/x)y=0 ----->Answer$$