# Separable partial differential equation

1. May 6, 2015

### whatisreality

1. The problem statement, all variables and given/known data
I have two equations.
cos(θ)wφ + sin(θ)wφ = 0 (1)
And
$\frac{w_φ}{r}$ + ∂wφ/∂r = 0 (2)
Find wφ, which is a function of both r and theta.

2. Relevant equations

3. The attempt at a solution
I end up with two equations, having integrated. wφ=$\frac{A}{sinθ}$ from (1), and wφ = Ar from (2). A is a constant of integration.
Is A the same in both equations? How do I combine the two solutions into one? I don't think you add them.

2. May 6, 2015

### 6c 6f 76 65

Have you typed the the first equation correct? Because if not $w_φ=0$, and for the rest: remember that when you differentiate / integrate with respect to $r$ you may have lost a function of θ.

3. May 6, 2015

### whatisreality

I know it isn't zero... But actually the first equation isn't correct, typed wrong.
Should be sin(θ) multiplied by the partial differential of wφ with respect to theta

4. May 6, 2015

### whatisreality

Did I differentiate instead of integrating? I checked the integration, can't find a mistake.

Last edited: May 6, 2015
5. May 6, 2015

### SammyS

Staff Emeritus
How about writing that first equation again in its corrected form, in its entirety .

6. May 6, 2015

### whatisreality

Sorry. Here it is:
cos(θ)wφ+sin(θ) $\frac{∂w_φ}{dθ}=0$.

7. May 6, 2015

### SammyS

Staff Emeritus
OK.

For your question regarding the constant of integration, A .

You would not expect the two constants of integration to be the same.

In general, the constant of integration you get by integrating over θ, can be a function of r , etc.