User:Egm6321.f12.team06.gu/report2

==Problem *2.2: Validation for Given L1-ODE-CC ==

Problem Statement

 * For the L1-ODE-CC
 * $$\displaystyle p'+p=x$$
 * Verify that
 * $$\displaystyle p(x)=k_1e^{-x}+x-1$$
 * is the solution for this equation.

Given

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$$\displaystyle p'+p=x$$
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 * (2.1)
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$$\displaystyle p(x)=k_1e^{-x}+x-1$$
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 * (2.2)
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Nomenclature

 * L1:First Order Linear Equation
 * ODE: Ordinary Differential Equation
 * CC: Constant Coefficients
 * $$\displaystyle p'=\frac{dp}{dx}$$

Solution

 * The first order derivative of Equation 2.2 can be expressed as:
 * $$\displaystyle p'(x)=(k_1e^{-x}+x-1)'=-k_1e^{-x}+1$$
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$$\displaystyle p'(x)=-k_1e^{-x}+1$$
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 * (2.3)
 * }
 * Thus, the left-hand side of Equation 2.1 becomes
 * $$\displaystyle p'+p=(-k_1e^{-x}+1)+(k_1e^{-x}+x-1)$$
 * $$\displaystyle =k_1e^{-x}+k_1e^{-x}+1-1+x$$
 * $$\displaystyle =x$$
 * Which is equal to the right-hand side of Equation 2.1.
 * Therefore, we can conclude that
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$$\displaystyle p(x)=k_1e^{-x}+x-1$$ is a solution for equation $$\displaystyle p'+p=x$$
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 * }

Problem Statement

 * For N1-ODE
 * $$\displaystyle M(x,y)+N(x,y)\,y'=0$$
 * Show that
 * a) This equation is affine in $$\displaystyle y'$$
 * b) This equation is in general an N1-ODE
 * c) This equation is not the most general N1-ODE. Give an example of a more general N1-ODE

Nomenclature

 * N1:Nonlinear First Order Equation
 * ODE: Ordinary Differential Equation
 * $$\displaystyle y'=\frac{dy}{dx}$$

Part a): Proving Affinity of Equation 3.1

 * Denote that


 * In general,to prove $$\displaystyle G(y',y,x)$$ is linear in $$\displaystyle y'$$
 * One just need to show that


 * Now we divide Equation 3.1 into two parts,
 * For the first part, since $$\displaystyle M(x,y)$$ is not a function of $$\displaystyle y'$$, one can just consider $$\displaystyle M(x,y)$$ as a constant with respect to $$\displaystyle y'$$
 * Clearly, for a constant $$\displaystyle M(x,y)$$
 * $$\displaystyle G_M(\alpha u+\beta v,y,x)=M(x,y)$$
 * $$\displaystyle \alpha\,G_M(u,y,x)+\beta\,G_M(v,y,x)=\alpha\,M(x,y)+\beta\,M(x,y)=(\alpha+\beta)\,M(x,y) \neq M(x,y),\forall \alpha,\beta\in \mathbb{R} $$
 * Therefore, the first part is nonlinear in $$\displaystyle y'$$ because
 * $$\displaystyle G_M(\alpha u+\beta v,y,x)\neq \alpha\,G_M(u,y,x)+\beta\,G_M(v,y,x)$$
 * For the second part, we have
 * $$\displaystyle G_N(\alpha u+\beta v,y,x)=N(x,y)\,(\alpha u+\beta v)=\alpha\,N(x,y)\,u+\beta\,N(x,y)\,v$$
 * While


 * $$\displaystyle \alpha\,G_N(u,y,x)+\beta\,G_N(v,y,x)=\alpha\,N(x,y)\,u+\beta\,N(x,y)\,v$$
 * Thus,


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$$\displaystyle G_N(\alpha u+\beta v,y,x)=\alpha\,G_N(u,y,x)+\beta\,G_N(v,y,x)$$
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 * (3.4)
 * }
 * In conclusion, we have proved that the second part of $$\displaystyle M(x,y)+N(x,y)\,y'$$ is linear in $$\displaystyle y'$$, while the first part is an intersection constant that is nonlinear in $$\displaystyle y'$$


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Therefore, Equation 3.1 is affine in $$\displaystyle y'$$
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Step 1: To prove Equation 3.1 is nonlinear

 * Denote that


 * In general, to prove $$\displaystyle G(y,x)$$ is nonlinear in $$\displaystyle y$$
 * One just need to show that


 * Substitute $$\displaystyle y$$ with $$\displaystyle \alpha u+\beta v$$, then Equation 3.2 becomes
 * $$ \displaystyle G(y',\alpha u+\beta v,x)=M(x,\alpha u+\beta v)+N(x,\alpha u+\beta v)\,\frac{d(\alpha u+\beta v)}{dx} $$
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$$ \displaystyle =M(x,\alpha u+\beta v)+\alpha N(x,\alpha u+\beta v)\,\frac{du}{dx}+\beta N(x,\alpha u+\beta v)\,\frac{dv}{dx} $$
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 * (3.7)
 * }
 * $$ \displaystyle \alpha G(y',u,x)+\beta G(y',v,x)= \alpha M(x,u)+\alpha N(x,u)\,\frac{du}{dx}+\beta M(x,v)+\beta N(x,v)\,\frac{dv}{dx} $$
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$$ \displaystyle =\alpha[M(x,u)+M(x,v)]+\alpha N(x,u)\,\frac{du}{dx}+\beta N(x,v)\,\frac{dv}{dx} $$
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 * (3.8)
 * }


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$$\displaystyle G(y',\alpha u+\beta v,x) \neq \alpha G(y',u,x)+\beta G(y',v,x)$$
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 * (3.9)
 * }
 * Therefore, Equation 3.1 is nonlinear

Step 2: To prove Equation 3.1 is First Order ODE

 * From (3.1), we can see that the highest order of derivative is y' ,which is the first order derivative of y with respect to x, thus it is a "First Order" equation;
 * Also, since $$\displaystyle y'=\frac{dy}{dx}$$, we know that "x" is the only independent variable for the derivative y', therefore this is an "Ordinary Differential Equation"


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In general, we can conclude that Equation 3.1 is in general a N1-ODE
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Part c): Example of More General N1-ODEs

 * General N1-ODE is expressed as
 * $$\displaystyle G(y',y,x)=0$$, where $$\displaystyle G$$ is a nonlinear function
 * Equation (3.1) is just one particular case of $$\displaystyle G(y',y,x)=0$$ that is affine in $$\displaystyle y'$$
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An example of a more general N1-ODE could be $$\displaystyle e^x+tan(y')+cos(x^3)+\frac{x^{17}}{14}+log\,y=0$$
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