User:Egm6321.f10.team6.lee/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).

Required
Solve the L2_ODE_VC when$$\begin{align} f(x)=0 \end{align}$$.

problem a
Using general trial solution,

hence,

Plug in eqn(2.4) to eqn(2.1). Then,

Then,

we can rewrite the eqn(2.5) in terms of x.

we can determine set of constants c and r from the eqn(2.6).

i.e.

problem b
using the same procedure to the problem a, let

hence,

Plug in eqn(2.4) to eqn(2.2). Then,

Then,

we can rewrite the eqn(2.9) in terms of x.

we can determine set of constants c and r from the eqn(2.10).

i.e.