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April 24th, 2017, 04:34 PM   #1
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N-th order Differential.Equation

...............This method does not contain ANALYTIC methods....!!!
Equation $\displaystyle \sum_{p=0}^{n}j_pY^{(n)}=j_n\frac{d^{n}Y}{dx^{n}}+ j_{n-1}\frac{d^{n-1}Y}{dx^{n-1}}+j_{n-2}\frac{d^{n-2}Y}{dx^{n-2}}+ ... .+j_2\frac{d^{2}Y}{dx^{2}}+ j_1\frac{dY}{dx}+y=0 $ $\displaystyle \; \;$ $\displaystyle {\{j,n\}}\in\{\Re,N\}$
$\displaystyle M(x,y)dx+N(x,y)dy=0 \; F(X)$ is solution $\displaystyle \rightarrow$ $\displaystyle \; dy/dx=-M/N=-\frac{\partial f / \partial x}{\partial f / \partial y}$ $\displaystyle \; \;$ $\displaystyle \begin{cases} \partial f / \partial x=\varphi M \\ \partial f / \partial y = \varphi N \end{cases}$ $\displaystyle \;$ so $\displaystyle \exists \varphi$ $\displaystyle \; $ integrating factor
Existence of integrating factor satisfy equation$\displaystyle M(x,y)dx+N(x,y)dy=0\;\; $$\displaystyle \: $$\displaystyle \frac{\delta Y}{M\delta x}=\frac{\delta Y}{N\delta y}\; \Rightarrow$$\displaystyle \exists \Large \Lambda_{j=0}^{i}$$\displaystyle [\ni_j(\delta y)]$$\displaystyle \rightarrowtail \delta^{c_{1}} \chi=\delta^{c_{2}}\chi$ $\displaystyle \; $ $\displaystyle \Rightarrow$ $\displaystyle \max[\Delta C]=2 $ or $\displaystyle \mu(x)=m(x)y'$$\displaystyle \;$ Defines Universal Structure of $\displaystyle \sum_{p=0}^{n}j_pY^{(n)}$ $\displaystyle \; $
Let $\displaystyle \phi$ be Universal Function Structure $\displaystyle \; $ $\displaystyle \Rightarrow \phi(\sum_{p=0}^{n}j_pY^{(n)})=\phi(\sum_{p=0}^{2} j_nY^{(n
)})=j_2y''+j_1y'+j_0y$
Let $\displaystyle \Psi(Y) $ formal function , Formal functions $\displaystyle \begin{cases} j_2z''+j_1z'+j_0z \\ j_2\mu''+j_1\mu'+j_0\mu \\ \end{cases}$ $\displaystyle \; \rightarrow $ $\displaystyle W(z,\mu)=e^{-\xi x}$ $\displaystyle \; $ and $\displaystyle \; $ $\displaystyle \Psi(Y)=\sqrt{e^{\epsilon}\theta}$ Wronskian shows \begin{cases} \Psi(j_2Y''+j_1Y'+j_0 Y)=\Psi(Y''-qY) \\ \Psi(Y)=e^{\mu x}A(x) \end{cases} $\displaystyle \; $ $\displaystyle Y''-qY=0 \equiv \Psi(Y)=\Psi(j_2Y''+j_1Y'+j_0 Y)$ $\displaystyle \;$ Formal Function is definable
We showed that Formal Function that satisfies $\displaystyle j_2 y''+j_1 y'+j_0 y=0$ is equaivalent to $\displaystyle y''-qy=0$ ...... $\displaystyle y=e^{rx}$
If $\displaystyle \delta^{0} \chi=\delta^{2}\chi$ $\displaystyle \; $ $\displaystyle \; \; \Psi(\sum_{p=0}^{n}j_pY^{(n)})\leftrightarrow jY^{(i)}-cY^{(i-2)}=0$ $\displaystyle \; \; \; $ Solution to differential equation $\displaystyle j_n y^{(n)}+j_{n-1}y^{(n-1)}+....+j_1y^{(1)} +y=0$ $\displaystyle \; $ is $\displaystyle \; $ $\displaystyle Y=e^{rx}\mu(x)$
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