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For the non-constant coefficients linear homogeneous ODE, there are only two solution methods: - If the ODE is like \(x^{2}y''+bxy'+cy=0\), then it can be fully solved: "Euler Equation" Method - Otherwise, need "Red of Order"

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The general form of 2nd order linear inhomogeneous ODE is \(\frac{d^{2}y}{dx^{2}}+p(x)\frac{dy}{dx}+q(x)y(x)=r(x)\), where \(p(x),q(x),r(x)\) are the function of \(x\).

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For 2nd Order Linear Inhomogeneous ODE, we need to find its complementary functions and its particular integrals. Getting its complementary functions by solving the homogeneous ODE constructed by replacing \(r(x)\) with 0 to get its complementary functions. Getting its particular integrals using the "Variation of Parameters" method.

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For a 2nd order linear inhomogeneous ODE \(\frac{d^{2}y}{dx^{2}}+p(x)\frac{dy}{dx}+q(x)y(x)=r(x)\) The particular integral of it is given by \(y_{p}(x)=u_{2}(x)\int^{x}\frac{u_{1}(\zeta)r(\zeta)}{W[u_{1}(\zeta),u_{2}(\zeta)]}d\zeta-u_{1}(x)\int^{x}\frac{u_{2}(\zeta)r(\zeta)}{W[u_{1}(\zeta),u_{2}(\zeta)]}d\zeta=\int^x \frac{\left|\begin{array}{cc}u_1(\zeta) & u_2(\zeta) \\u_1(x) & u_2(x)\end{array}\right|}{W\left[u_1(\zeta), u_2(\zeta)\right]} r(\zeta) d \zeta\) where: - \(\displaystyle W[u_{1}(x),u_{2}(x)]=\begin{vmatrix}u_{1}(x) & u_{2}(x) \\ u_{1}'(x) & u_{2}'(x)\end{vmatrix}=u_{1}(x)u_{2}'(x)-u_{2}(x)u_{1}'(x)\). - \(u_{1}(x), u_{2}(x)\) is the complementary functions of the inhomogeneous ODE. - \(r(x)\) is RHS of the inhomogeneous ODE.

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In electrostatics, we ask \(\vec{E}\) to be independent of time, but it can change with position. \(\nabla\cdot\vec{E}=\frac{\rho}{\varepsilon_{0}}\) We also ask \(\vec{B}\) to be independent of time. \(\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}=0\) - This equation is used to decide whether an electric field can be an el