User:EGM6321.f10.team6.yoshioka/HW6

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

Given
The non-homogeneous Legendre Equation with n=1:

And the 1st homogeneous solution is given by:

Find
The solution of the Legendre equation using the direct method.

Solution
From (1) 29-1, let the solution be in the form:

And the derivatives of 3.3:

Substituting equations 3.3, 3.4 and 3.5 into 3.1 we have:

By rearranging the equation 3.6:

Calculating the derivatives of the 1st homogeneous solution (equation 3.2):

Substituting equations 3.2, 3.8 and 3.9 into 3.7:

The equation 3.10 results in:

By applying the reduction of order method as presented in Meeting 5, z(x) can be defined as:

Substituting the equation 3.12 into 3.11:

Rearranging the equation 3.13:

Since the ODE is in the form of:

Calculating the integrating factor by using (1) 10-2

From (6) 10-3 the solution of z(x) is given by:

And the solution of U(x) is given by:

Finally the solution of y(x) is calculated by using (2) 29-3:

The final solution of y(x) is given by:

Given
The spherical coordinates:

Find
The infinitesimal segment ds and the Laplace equation.

Solution
The infinitesimal segment ds in the 3 dimensional space is given by:

Where:

Therefore applying the Einstein summation in the equation 4.5:

The general equation to find $$d{{x}_{i}}$$ is given by:

Calculating $$d{{x}_{1}}$$, $$d{{x}_{2}}$$ and $$d{{x}_{3}}$$ from equation 4.7.

Calculating the squares of equation 4.10, 4.13 and 4.16:

Substituting equations 4.18, 4.20 and 4.22 into 4.6 and grouping them in terms of $$dr$$, $$d\theta $$ and $$d\varphi $$:

Using the identities of $${{\cos }^{2}}(\varphi )$$ and $${{\sin }^{2}}(\varphi )$$ into 4.23:

Finally, the the infinitesimal segment ds in the 3 dimensional space can be defined by:

The Laplace equation in the spherical coordinates is defined by:

Where:

Substituting 4.28, 4.29 and 4.30 into 4.27:

Therefore $$\psi $$ can be calculated by:

Finally the Laplace equation for the spherical coordinates id defined by: