Using the Frobenius method on a 2D Laplace

[jkthejetplane]

Homework Statement

I have some supplemental hmwk that I need to use a method I am not familiar with and I am not sure I have applied it correctly bc if i solve for the missing variable i will not achieve the right answer.

Relevant Equations

Frobenius method of solving ode

 

[vela]

The problem is you're using the symbol $\ell$ in two different ways: (1) as part of the constant that appears on the righthand side of the differential equation and (2) as the index of the summation.

 

[vanhees71]

Hint:
$$\frac{\mathrm{d}}{\mathrm{d} r} \left (r^2 \frac{\mathrm{d} R}{\mathrm{d} r} \right)=r \frac{\mathrm{d}^2}{\mathrm{d} r^2} (r R)$$
helps to make the equation simpler.

Then you have to choose another summation variable in your Frobenius ansatz than $l$, e.g.,
$$R(r)=\sum_{j=0}^{\infty} a_j r^{j+\lambda},$$
and you can assume $a_0 \neq 0$ for uniqueness. Then plug this ansatz into your ODE and determine first $\lambda$ and then the $a_j$ by comparing the coefficients in the generalized power series on both sides of your equation.

 

[jkthejetplane]

Hint:
$$\frac{\mathrm{d}}{\mathrm{d} r} \left (r^2 \frac{\mathrm{d} R}{\mathrm{d} r} \right)=r \frac{\mathrm{d}^2}{\mathrm{d} r^2} (r R)$$
helps to make the equation simpler.

Then you have to choose another summation variable in your Frobenius ansatz than $l$, e.g.,
$$R(r)=\sum_{j=0}^{\infty} a_j r^{j+\lambda},$$
and you can assume $a_0 \neq 0$ for uniqueness. Then plug this ansatz into your ODE and determine first $\lambda$ and then the $a_j$ by comparing the coefficients in the generalized power series on both sides of your equation.

Ok so as far as your hint goes, I should be using that whole expression by getting it equal to zero so I have a y" term or that I should only use the right side equal to 0?

 

[jkthejetplane]

The problem is you're using the symbol $\ell$ in two different ways: (1) as part of the constant that appears on the righthand side of the differential equation and (2) as the index of the summation.

Oh yeah i see that now. I got caught up in our notation and other notation when looking up the method. I'll try again using the standard n

 

[vanhees71]

Ok so as far as your hint goes, I should be using that whole expression by getting it equal to zero so I have a y" term or that I should only use the right side equal to 0?

Just plug in the ansatz into the equation and compare the coefficients of the series. You'll get an equation for $\lambda$ and then a recursion relation for the $a_j$.

 

[jkthejetplane]

Update: I got it all correct after just trying again without the confusion of l terms. Did it a couple ways to make sure. Thank you