User:Egm6321.f10.team6.Kim.MK/hw6

=Problem 3 - Solving the Legendre Equation Using the Direct Method =

Find
The complete solution $$y(x)$$ using the direct method

Solution
We can make (3.1) as like (3.3) form,

1st homogeneous solutions is (3.2) $$\begin{align}{{u}_{1}}(x)=x\end{align}$$ with $${{a}_{1}}(x)=\frac{-2x}{1-{{x}^{2}}}$$

The integrating factor is

2nd homogeneous solution can be obtained as

Then

Thus,

The particular solution can be obtained as

with $$\begin{align}f(x)=\frac{1}\end{align}$$

Then,

$$\begin{align} \end{align}$$

Therefore, the complete solution can be written as

Note
The integral of $$\begin{align}y_{p}\end{align}$$ can be evaluated via Wolfram Alpha

Given
The equations for spherical coordinates with Math/Phys. convention are given by:

Solution of Problem 4 and definition of $$\theta $$ measured clockwisely from the North Pole

Find
The expression of Lapalcian

Solution
Eq. (6.2) can be written as

Replace (6.4)in the Laplacian obtained in Problem 4,

Thus,

So, we can get