User:EGM6321.f12.team7.Zhou/R2

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Problem 1: Verify the Homogeous Solutions
Report problem 2.1 from

Given: Two homogeous solutions
The Legendre differential equation is given by,

When the $$n=1$$, we have,

We also have two linearly-independent solutions, which are given by,

Find
Verify that,$$ L_0(Y^2_H(x))= L_0(Y^1_H(x))=0$$.

Solution

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This solution was prepared without referring to previous solutions.
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For $$ L_0(Y^1_H(x))$$, we have,

Then the $$\displaystyle L_0(Y^1_H(x))$$ comes to,

For $$\displaystyle L_0(Y^2_H(x))$$, we have,

So the $$\displaystyle L_0(Y^2_H(x))$$ changes into,

So we have,

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Problem 8: Solving equations for integrating factor
Report problem 2.8 from lecture notes

Given:Euler Integrating Factor Method
When we find a N1-ODE can not fit the first exactness condition but fail on the second one, we can use Euler Intergrating Factor to make the equation exact, which is given by,

Apply the second exactness condition, we have,

Where,

$$h_{x}:=\frac{\partial h}{\partial x},$$ $$h_{y}:=\frac{\partial h}{\partial y},$$ $$N_{x}:=\frac{\partial N}{\partial x},$$ $$M_{y}:=\frac{\partial M}{\partial y}$$

Solution

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This solution was prepared without referring to previous solutions. In ($$), since $$h$$ is nonlinear and we treat x and y as two independent variables, so the equation turns into a $$N1-PDE$$, so we have to solve this problem in two dimensional domain. What's more, the coefficients of $$h$$, $$h_{x}$$ and $$h_{y}$$ have varying coefficients ($$N_{x}, $$$$M_{y}, $$$$M, $$$$N$$). Hence it is not easy to solve this equation. And that's also the reason to make assumpetion that $$h$$ is only a founction of $$x$$ or $$y$$.
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