# Thread: solving 2nd order PDE

1. ## solving 2nd order PDE

Need to solve for dx/ds in the following equation, keeping dy/ds.

d^2x/ds^2 - (2/y)(dx/ds)(dy/ds) = 0

I can just rearrange to get:

dx/ds = (y/2)(ds/dy)(d^2x/ds^2)

But, this is not clean to use for some later calculations.

Is there any way to solve for dx/ds by integration?I would like an expression with dx/ds = f(y,s)

2. First off, this is and ODE since there are only ordinary derivatives here. If you write your ODE as

$\dfrac{\dfrac{d^2x}{ds^2}}{\dfrac{d x}{d s} }= 2 \dfrac{\dfrac{dy}{ds}}{y}$

You can integrate giving

$\ln \dfrac{d x}{d s}} = 2 \ln y + \ln c$

or

$\dfrac{dx}{ds} = c y^2$