# Differential Equations

r" = 1/r^2

## Homework Equations

A friend gave this to me, he was just wondering how we'd approach it.

## The Attempt at a Solution

I don't think the equation is linear, so I don't know how to approach it. My friend suggested integrating both sides with respect to r, but we don't know if that's legal because we don't know what r might be a function of.

hunt_mat
Homework Helper
Multiple both sides by r' and that will give you the first integral.

Multiple both sides by r' and that will give you the first integral.

How's does multiplying by a derivative give me an integral?

hunt_mat
Homework Helper
As
$$r''=\frac{1}{r^{2}}$$
Multiply both sides by r' to obtain:
$$r'r''=\frac{r'}{r^{2}}\Rightarrow\left(\frac{(r')^{2}}{2}\right) '=\left( -\frac{1}{r}\right) '$$
Integrate easily from here.

HallsofIvy
Homework Helper
Did you try it? What is the derivative of $(r')^2$? So what is the integral of $r' r''$? What is the integral of $dr/r^2$?

To hunt_mat:

That's a very nice trick! I got stuck on the next step, integrating (r')2, but I'll try to work that out with my friend before asking again.

Thanks very much!

hunt_mat
Homework Helper
It's a standard trick, become familier with it.