Second order differential equation with non constant coefficients.

• Sep 8th 2012, 05:16 AM
Second order differential equation with non constant coefficients.
Hi all,

While solving some physics problem I came across a differential equation.

$\displaystyle \frac{\mathrm{d}^2y}{\mathrm{d}x^2}+c_1x^2y=c_2y$

Where all the constants $\displaystyle c_1$ and $\displaystyle c_2$ are positive. I need a solution for this equation. Please indicate me how to go about this. It will be helpful if the solution is in closed form. It might be helpful to know that, the solution to,

$\displaystyle \frac{\mathrm{d}^2y}{\mathrm{d}x^2}-c_1x^2y=c_2y$

is a product of hermite polynomials with a Gaussian envelope.

Thanking you.
Shantanu.
• Sep 8th 2012, 05:44 AM
HallsofIvy
Re: Second order differential equation with non constant coefficients.
In general, solutions to even linear differential equations, with variable coefficients, cannot be written in terms of "elementary" functions. If you already know that solutions to the second equation are "products" of Hermitian functions, why not try writing the solutions to the first equation that way with argument ix rather than x? Failing that, some sort of series solution would be called for.