User:Egm6321.f12.team7.parikh.a/R6.11

=R*6.11=

Problem: Find the 2nd homogeneous solution of the Legendre equation
Problem R*6.11 from lecture notes section 37.

Given: The 1st homogeneous solution
For Legendre's equation, given by:

The first homogeneous solution when $$n=2$$ is

Find: The 2nd homogeneous solution
Show that

is the second homogeneous solution using variation of parameters.

Solution: Reduction of order formula

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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First, note the typo in the problem definition. The 1st homogeneous solution should be

The variation of parameters method is only used to find the particular solution to a non-homogeneous differential equation once the homogeneous solutions are already known. However, we are trying to find the second homogeneous solution to the homogeneous Legendre equation, therefore the reduction of order formula will be used.

First, Legendre's equation is rearranged.

Now we can evaluate the inner integral of the reduction of order formula. Setting $$$$

From this it follows

Therefore the reduction of order formula now becomes

Using the WolframAlpha integrator, the right hand side evaluates to

After rearranging and simplifying, the 2nd homogeneous equation as shown in the problem statement is found.

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