Seperaion of Variables (PDE's)

[twoscoops]

Homework Statement

Look for a seperable solution T(r,θ) = R(r)Θ(θ) and derive equations for R(r) and Θ(θ) choosing a separation constant that gives sinusoidal solutions for Θ(θ). Write down a general solution for Θ(θ) and show the equation for R(r) has solutions of the form R(r)=r^p.

Homework Equations

p is just a random letter which has no real meaning.

The Attempt at a Solution

I can get the solution of Θ(θ)=Asinkθ+Bcoskθ using k^2 as the separation constant. That leaves me with r²R''+ rR' = k²R but I am not sure what solution that gives, likewise for the general solution for Θ(θ). Any help will be much appreciated. Thanks

 

[jackmell]

Homework Statement

Look for a seperable solution T(r,θ) = R(r)Θ(θ) and derive equations for R(r) and Θ(θ) choosing a separation constant that gives sinusoidal solutions for Θ(θ). Write down a general solution for Θ(θ) and show the equation for R(r) has solutions of the form R(r)=r^p.

When you are required to "show" a function satisfies a DE, then just substitute that function into the DE and see if it satisfies it. So you're asked to "show" that R(r)=r^p satisfies the DE:

$$r^2 R''+rR'=k^2 R$$

so when you substitute R(r)=r^p into that, what must the relationship between p and k be so that it satisfies the DE?

 

[yungman]

When you are required to "show" a function satisfies a DE, then just substitute that function into the DE and see if it satisfies it. So you're asked to "show" that R(r)=r^p satisfies the DE:

$$r^2 R''+rR'=k^2 R$$

so when you substitute R(r)=r^p into that, what must the relationship between p and k be so that it satisfies the DE?

$r^2 R''+rR' - k^2 R= 0$ is Euler equation and solution is $r^n \hbox { or } r^p$ what ever which way you want to call it.

 

[twoscoops]

okay thanks for clearing that up, but what about the general solution for Θ(θ)? still not sure on that...

 

[vela]

You said you already found it in your original post. What specifically are you stuck on?