# Power law trick DE

1. Nov 28, 2007

### Anito

1. The problem statement, all variables and given/known data

I have no idea how to solve this differential equasion:(d^2y/ds^2)=L^2/(y^3)

where L is constant. It looks like a inhomogenius DE but what should I do with y^3?

Last edited by a moderator: Jul 15, 2014
2. Nov 28, 2007

### Dick

Try looking for a power law solution y=As^k.

3. Nov 28, 2007

### HallsofIvy

Staff Emeritus
Dick, it that were $y^2$ on the right side that would work (it would be an "Euler-type" equation) but I don't think it works here. Since the independent variable, s, does not appear explicitely, I would try "quadrature".

Let v= dy/dt so $d^2y/dt^2= dv/dt= (dv/dy)(dy/dt)= v dv/dy$. The equation becomes v dv/dy= L/y3. vdv= Ly-3dy. Integrate that to get (1/2)v2= (-L/2)y-2+ C. Since v= dy/dt, that is
$$\frac{dy}{dt}= \sqrt{C- Ly^{-2}}$$

4. Nov 28, 2007

### Dick

Thanks, Halls. The power law trick does give you a particular solution proportional to s^(1/2), but that way you get a more general solution.

Last edited: Nov 28, 2007