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Question

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[Using "Varient of Parameters" and show all forms of the formula]

Answer

\(y_{p}(x)=u_{2}(x)\int^{x}\frac{u_{1}(\zeta)r(\zeta)}{W[u_{1}(\zeta),u_{2}(\zeta)]}d\zeta-u_{1}(x)\int^{x}\frac{u_{2}(\zeta)r(\zeta)}{W[u_{1}(\zeta),u_{2}(\zeta)]}d\zeta=\int^x \frac{\left|\begin{array}{cc}u_1(\zeta) & u_2(\zeta) \\u_1(x) & u_2(x)\end{array}\right|}{W\left[u_1(\zeta), u_2(\zeta)\right]} r(\zeta) d \zeta\)

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.


Question

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[Using "Varient of Parameters" and show all forms of the formula]

Answer
?

Question

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[Using "Varient of Parameters" and show all forms of the formula]

Answer

\(y_{p}(x)=u_{2}(x)\int^{x}\frac{u_{1}(\zeta)r(\zeta)}{W[u_{1}(\zeta),u_{2}(\zeta)]}d\zeta-u_{1}(x)\int^{x}\frac{u_{2}(\zeta)r(\zeta)}{W[u_{1}(\zeta),u_{2}(\zeta)]}d\zeta=\int^x \frac{\left|\begin{array}{cc}u_1(\zeta) & u_2(\zeta) \\u_1(x) & u_2(x)\end{array}\right|}{W\left[u_1(\zeta), u_2(\zeta)\right]} r(\zeta) d \zeta\)

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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Variation of Parameters for getting the particular integral of a 2nd order linear homogeneous ODE
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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