User:Egm6321.f12.team6.bang/report4/problem*4.1

=Report 4= ==Problem *4.1: Validation of Exactness Condition for L2-ODE-VC ==

Problem Statement

 * Verify the Exactness of $$ \displaystyle \sqrt{x}y'' + 2xy' + 3y = 0 $$

Given

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$$ \displaystyle \sqrt{x}y'' + 2xy' + 3y = 0 $$
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 * (6.1)
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 * 1st Exactness condition:
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$$ \displaystyle G(x,y,y',y) = g(x,y,p) + f(x,y,p)y = 0 $$
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 * (6.2)
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 * 2nd Exactness Condition:
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$$ \displaystyle f_{xx} + 2pf_{xy} + p^2f_{yy} = g_{xy} + pg_{yp} - g_{y} $$
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 * (6.3)
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$$ \displaystyle f_{xp} + pf_{yp} + 2f_{y} = g_{pp} $$
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 * (6.4)
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Nomenclature

 * L2-ODE - Linear Second Order Differential Equation
 * $$ \displaystyle y' = p $$
 * $$ \displaystyle f_{xx} = \frac {\partial^2 {f}}{\partial{x^2}}; f_{xy} = \frac {\partial^2 {f}}{\partial{x} \partial{y}}; f_{xp} = \frac {\partial^2 {f}}{\partial {x} \partial {p}} = \frac {\partial^2 {f}}{\partial {x} \partial {y'}}$$

Solution

 * Above Equation 6.1 can be expressed as following.
 * $$ \displaystyle \sqrt{x}y'' + 2xp + 3y = 0 $$
 * $$ \displaystyle \underbrace {2xp + 3y}_{\displaystyle g(x,y,p)} + \underbrace {\sqrt{x}}_{\displaystyle f(x,y,p)}y'' = 0 $$
 * Comparing this with Equation 6.2,
 * $$ \displaystyle G(x,y,y',y) = g(x,y,p) + f(x,y,p)y = 0 $$
 * Therefore, this equation passed the first condition of exactness.
 * In order to evaluate 2nd exactness condition, we have to take many different partial differentiation of $$ \displaystyle g(x,y,p) $$, and $$ \displaystyle f(x,y,p) $$.
 * $$ \displaystyle f = \sqrt {x} = x^{\frac {1}{2}} $$
 * $$ \displaystyle f_{x} = \frac{1}{2}x^{-\frac{1}{2}}; f_{y} = 0 $$
 * $$ \displaystyle f_{xx} = -\frac{1}{4}x^{-\frac{3}{2}}; f_{xy} = 0; f_{xp} = 0; f_{yy} = 0; f_{yp} = 0 $$
 * $$ \displaystyle g = 2xp + 3y$$
 * $$ \displaystyle g_{x} = 2p; g_{y} = 3; g_{p} = 2x $$
 * $$ \displaystyle g_{xp} = 2; g_{yp} = 0; g_{pp} = 0$$
 * If we put all corresponding terms into Equation 6.3 and Equation 6.4, then we get two sets of equations.
 * $$ \displaystyle -\frac{1}{4}x^{-\frac{3}{2}} + 0 + 0 \; ? \; 2 + 0 - 3 $$
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$$ \displaystyle -\frac{1}{4}x^{-\frac{3}{2}} \neq -1 $$
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 * (6.5)
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 * $$ \displaystyle 0 + 0 + 0 \; ? \; 0$$
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$$ \displaystyle 0 = 0 $$
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 * (6.6)
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 * Even though Equation 6.4 satisfies as shown from Equation 6.6, Equation 6.3 does not satisfy as shown from Equation 6.5. Therefore, the given equation fails the second condition of exactness.
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Since the equation fails to satisfy one of conditions, this equation is not in a form of exact L2-ODE.
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