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order linear inhomogeneous ODE \(\frac{d^{2}y}{dx^{2}}+p\frac{dy}{dx}+qy(x)=f(x)\) where \(f(x)\) is a periodic function that given the period \(L_{0}\). To find the ODE's particular integral, <span>1. Calculate the Fourier series of the function \(f(x)\) 2. Try ansatz and using the period \(L_{0}\): \(\displaystyle y(x)=\frac{a_{0}}{2}+\sum\limits^{\infty}_{n=1}a_{n}\cos\left(\frac{n\pi x}{L_{0}}\right)+b_{n}\sin\left(\frac{n\pi x}{L_{0}}\right)\). 3. Yield the derivatives of \(y(x)\) and plug them into the original ODE. 4. Comparing coefficients in front of 1, cosine, sine, respectively, write out the equation system about the ODE. 5. Solve the equation system and write out the particular integral. <span>

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Four things form the basic structure of any PDE problem: - TF: Target function(Function of interests \(\phi(x,t)\)) - DOV: Domain of Validity, such as \(x_{1}<x<x_{2}\) and \(t_{1}<t<t_{2}\). - PDE: Partial Differential Equation - BC: Boundary condition

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The general form of \(2^{nd}\) order linear homogeneous PDE is \(\displaystyle A(x)\phi_{xx}+B(x)\phi_{xy}+C(x)\phi_{yy}+D(x)\phi_{x}+E(x)\phi_{y}+F(x)\phi=0\) where \(A(x),B(x),C(x),D(x),E(x),F(x)\) are the function of \(x\), and \(\phi_{xx}\) stands for \(\displaystyle\frac{\partial^{2}\phi}{\partial x^{2}}\), \(\phi_{x}\) stands for \(\displaystyle\frac{\partial\phi}{\partial x}\), etc.

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From the general form of the linear homogeneous PDE, we can divide all PDE into 3 types: 1. \(\Delta\equiv B^{2}-4AC>0\Rightarrow\) Hyperbolic PDE(Wave Equation): \(\phi_{tt}=c\phi_{xx}\) 2. \(\Delta\equiv B^{2}-4AC=0\Rightarrow\) Parabolic PDE(Heat/Diffusion Equation): \(\phi_{t}=k\phi_{xx}\) 3. \(\Delta\equiv B^{2}-4AC<0\Rightarrow\) Elliptical PDE(Laplace Equation): \(\phi_{xx}+c\phi_{yy}=0\)