User:EGM6321.f12.team4.Rui.C/Homework6 R6.12

Statement
Consider the Nonhomogeneous Legendre equation:

Given the 1st homogeneous solution:

Find the final solution y(x) by variation of parameters.

Solution
Set $$y(x)=U(x)u_1(x)=U(x)x$$ and substitute it into the Legendre equation;

Since coefficient for $$U(x)$$ equals to 0, we can assume $$Z(x)=U'(x)$$. Then (6.12.5) becomes

It is obvious that the 1st exactness condition is satisfied:

Since

The second exactness condition does not hold. We have to use IFM to solve this linear L2-ODE-VC. Assume $$h(x,y)$$ ,

Set:

Assume $$h=h(x)$$ ,

Then:

So

By substituting (6.12.13) into (6.12.9), we can have:

where $$k=C_2/C_1$$. Then U can be found by integrating Z with respect to x:

If we set $$k'=-k$$, then:

By substituting $$U(x)$$ back, y can be written as:

This is the final result of the Legendre equation, in which the first term is the particular root, while the second and the third therms are the 2nd and the 1st homogeneous roots respectively.

Author and References

 * Solved and Typed by -- Rui Che