User:Egm6321.f11.team4.YuChen/HW2

=Problem 2.1-Verify the Given Equation=

Given
The Legendre differential equation:

When $$\displaystyle n=1 $$, it becomes:

The two homogeneous solutions for equation (1.2) are:

Find
Verify that when $$\displaystyle n=1 $$,

is true.

Solution
Substituting $$\displaystyle y $$ for $$\displaystyle x $$ in equation (1.2):

Substituting $$\displaystyle y $$ for $$\displaystyle \frac{x}{2}ln(\frac{1+x}{1-x}) $$ in equation (1.2):

Derive $$\displaystyle y^2_H = \frac{x}{2}ln(\frac{1+x}{1-x}) $$ to get its first and second order derivitives:

Substituting in equation (1.7):

So equation (1.5)is true.

=Problem 2.2 - Verify the Solution of an Equation=

Given
A differential equation:

Find
is the solution to equation (2.1).

Solution
Substituting $$\displaystyle p' $$ for (2.3) in equation (2.1):

End of proof.

=Problem 2.6-Show Linearly Independence=

Given
The two homogeneous solutions for equation (1.2) in problem 1.

Find
Show that the two solutions a linearly independent, and plot them.

Plotting
To make sense $$\displaystyle x $$ must be defined on the interval $$\displaystyle (-1,1) $$.

Plot $$\displaystyle y_{H}^{1} $$ and $$\displaystyle y_{H}^{2} $$. Matlab Code:



Proof of Linearly Independence
To prove linearly independence of $$\displaystyle y_{H}^{1} $$ and $$\displaystyle y_{H}^{2} $$ is to show that $$\displaystyle \forall \alpha \in \mathbb R, y_{H}^{1} \ne \alpha y_{H}^{2}$$.

That is $$\displaystyle \exists \hat x $$ such that $$\displaystyle y_{H}^{1}(\hat x) \ne \alpha y_{H}^{2}(\hat x) $$.

Take $$\displaystyle \alpha = 0 $$ for example.When $$\displaystyle \alpha = 0 $$, $$\displaystyle y_{H}^{1} = \alpha y_{H}^{2} $$ yields $$\displaystyle x=0 $$.

But when $$\displaystyle x=\frac{1}{2} $$, $$\displaystyle y_{H}^{1} \ne \alpha y_{H}^{2} $$.

So $$\displaystyle \forall \alpha \in \mathbb R, y_{H}^{1} \ne \alpha y_{H}^{2}$$.