Separable partial differential equation

[whatisreality]

Homework Statement

I have two equations.
cos(θ)w_φ + sin(θ)w_φ = 0 (1)
And
$\frac{w_φ}{r}$ + ∂w_φ/∂r = 0 (2)
Find w_φ, which is a function of both r and theta.

Homework Equations

The Attempt at a Solution

I end up with two equations, having integrated. w_φ=$\frac{A}{sinθ}$ from (1), and w_φ = 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.

 

[6c 6f 76 65]

Homework Statement

I have two equations.
cos(θ)w_φ + sin(θ)w_φ = 0 (1)
And
$\frac{w_φ}{r}$ + ∂w_φ/∂r = 0 (2)
Find w_φ, which is a function of both r and theta.

Homework Equations

The Attempt at a Solution

I end up with two equations, having integrated. w_φ=$\frac{A}{sinθ}$ from (1), and w_φ = 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.

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.

 

[whatisreality]

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.

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 w_φ with respect to theta

 

[whatisreality]

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

 

[SammyS]

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 w_φ with respect to theta

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

 

[whatisreality]

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

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

 

[SammyS]

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.