# Solve the following symetrical differential equation

Recently I encountered a differential equation which is as follows:

$\frac{d^3y}{dx^3} + x^3 \frac{d^2y}{dx^2} + 3x^2 \frac{dy}{dx} + 6xy + 6 = 0$

I couldn't solve it because we were not taught how to solve differential equations of higher orders.

My efforts: By looking at it I found the differential equation inform of symmetrical differential equation which is as follows:

$\frac{d^3y}{dx^3} + t \frac{d^2y}{dx^2} + \frac{dt}{dx} \frac{dy}{dx} + \frac{d^2t}{dx^2}y + \frac{d^3t}{dx^3} = 0$ where $t$ is a function of $x$. Then we have to solve it.

How to solve such sums ? Is there a general method ? As I don't know the method, few hints will be appreciated.

• Check the writing of your equation. There is probably a typo. As it is presently written, solving it requires too high level knowlege in the field of special functions. I bet that $x^4$ is missing in coefficient to the first term. – JJacquelin Apr 25 '16 at 13:57