Solving a first order differential equation

[stephen cripps]

Homework Statement

We have the equation
$(\frac{dr}{ds})^2+(\frac{l}{r})^2=1$
and want to solve to get $r=\sqrt{l^2+(s-s_0)^2}$

Homework Equations

The Attempt at a Solution

I have worked backwards, plugging in the solution to prove that it is correct, but the closest I have gotten to actually finding the solution without using r is: $\frac{dr}{ds}=\frac{\sqrt{r^2-l^2}}{r}$

Can anyone help with where to go from here?

 

[Geofleur]

What you have is a good start. Can you rewrite the equation just a little more so that only $r$ and $dr$ show up on one side and only $ds$ shows up on the other?

 

[BvU]

Hello Stephen,

I take it you have seen the solution satisfies the differential equation ?

And doesn't the solution remind you of good old Pythagoras ?

 

[Mark44]

Homework Statement

We have the equation
$(\frac{dr}{ds})^2+(\frac{l}{r})^2=1$
and want to solve to get $r=\sqrt{l^2+(s-s_0)^2}$

Homework Equations

The Attempt at a Solution

I have worked backwards, plugging in the solution to prove that it is correct, but the closest I have gotten to actually finding the solution without using r is: $\frac{dr}{ds}=\frac{\sqrt{r^2-l^2}}{r}$

Can anyone help with where to go from here?

Some differential equation problems take the form of "show that this equation is a solution of the differential equation ..." Other differential equation problems ask you to solve a given DE, and don't provide the solution. Your problem appears to be the latter type.

To start, note that $(\frac{dr}{ds})^2+(\frac{l}{r})^2=1$ can be rewritten as $\frac{dr}{ds} = \pm \sqrt{1 - (\frac{l}{r})^2}$

 

[stephen cripps]

What you have is a good start. Can you rewrite the equation just a little more so that only $r$ and $dr$ show up on one side and only $ds$ shows up on the other?

This is the part of the problem I'm having trouble with

 

[Mark44]

This is the part of the problem I'm having trouble with

What is the trouble you're having?
Starting from $\frac{dr}{ds} = \pm \sqrt{1 - (\frac{l}{r})^2}$, separate the variables by dividing both sides by $\sqrt{1 - (\frac{l}{r})^2}$, and multiplying both sides by ds. You will need to handle the + and - cases with an equation for each.

 

[stephen cripps]

Oh yeah, I have it now. I was being stupid.