User:EGM6321.f10.team6.cook/hw6

=Problem 1 - Legendre's Equation =

Problem Statement
Legendre's Equation is, for n=0,1,2,...

when n=2,

Required
If we were given the 1st homogeneous solution,

Show that the 2nd homogeneous solution is

Solution
For the general homogeneous L2_ODE_VC,

let's put the solution like eqn(1.6) to find out the 2nd homogeneous solution.

Then,

Plug in to the eqn(1.5),

So, we can get L2_ODE_VC with missing dep. var. U. To reduce order, let

then,

integrating factor for this ODE would be,

multiply eqn(1.11) to eqn(1.10) on both side.

Therefore, from eqn(1.6),

So we can figure out what the 2nd homogeneous solution is if we have the 1st homogeneous solution.

Let's rewrite the given problem eqn(1.2).

From the problem statement, we have the 1st homogeneous solution, eqn(1.2).

Let's find the integrating factor first. from eqn(1.11),

Hence,

This is the same to the eqn(1.4).

Problem Statement
The Legendre equation that arises as a result of separation of variables may be written as :

One set of solutions to this equation are the Legendre polynomials of the first kind which can be expressed generally as :

Required
Show that for n=0 to n=4 in equation 9.2 are solutions to the Legendre equation in 9.1.

Verification of n=0
By substituting n=0 into equation 9.2 we obtain the following solution to the Legendre equation:

In order to verify that this is a solution we must back substitute 9.3 into 9.1. If we do so we obtain the following:

Clearly for n=0 equation 9.2 provides a solution to the Legendre Equation.

Verification of n=1
By substituting n=1 into equation 9.2 we obtain the following solution to the Legendre equation:

In order to verify that this is a solution we must back substitute 9.6 into 9.1. If we do so we obtain the following:

For n=1 equation 9.2 provides a solution to the Legendre Equation.

Verification of n=2
By substituting n=2 into equation 9.2 we obtain the following solution to the Legendre equation:

In order to verify that this is a solution we must back substitute 9.10 into 9.1. If we do so we obtain the following:

For n=2 equation 9.2 provides a solution to the Legendre Equation.

Verification of n=3
By substituting n=3 into equation 9.2 we obtain the following solution to the Legendre equation:

In order to verify that this is a solution we must back substitute 9.14 into 9.1. If we do so we obtain the following:

For n=3 equation 9.2 provides a solution to the Legendre Equation.

Verification of n=4
By substituting n=4 into equation 9.2 we obtain the following solution to the Legendre equation:

In order to verify that this is a solution we must back substitute 9.18 into 9.1. If we do so we obtain the following:

For n=4 equation 9.2 provides a solution to the Legendre Equation.