# Not so obvious 4th order DE

## Homework Statement

solve the given differential equation:

xy^(4) + 6y'" = 0

## Homework Equations

No idea, I need something to get me started

## The Attempt at a Solution

auxiliary equation:

m^4 + 6m^3 = 0

m^3(m+6)=0
3 roots with m = 0 and the other root is m = -6

so I got y = C1 + C2lnx + C3(lnx)^2 + C4x^-6

but, it is not the solution. Your help is greatly appreciated. Thank You.

cristo
Staff Emeritus
I don't quite know what you're doing here. The method you are using here seems to be the one that you use for a second order equation, however this is an equation of 4th order.

Take a look at the equation and try and spot a solution. In particular, you have a fourth derivative multiplied by x, and a third order derivative. Now, you know a function whose fourth derivative multiplied by x is equal to its third derivative (modulo constants). The function is xn. So, try the general solution y=xn. Plugging this into the ODE will give you a condition on n, which will enable you to obtain the particular solution.

HallsofIvy
can you give me hint on how to solve an equation that involves a particular solution? Thank you 