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The "Reduce of Order" Method for non-constant-coefficients 2nd order linear homogeneous ODE

About most of non-constant coefficients 2nd linear homogeneous ODE


there is NO universal recipe. In general, it can only tried & error.
However, if we find 1 solution, there exist a recipe to find the 2nd:
1. Assume we have 1 solution: \(y_{1}(x)\). Propose ansatz: \(y_{2}(x)=V(x)y_{1}(x)\), with some unknow function \(V(x)\).
2. Derivative \(y''\), \(y'\) terms, plug them to the original ODE.
3. Let \(\displaystyle w= \frac{dV}{dx}\) and reduce it to the first order equation.\(a_{2}y_{1} \frac{dw}{dx}+(2a_{2}y_{1}'+a_{1}y_{1})w=0\)
4. Solve it and then we can get \(\displaystyle V(x)=\int e^{-\int(2y_{1}'/y_{1}+a_{1}/a_{2})dx}dx+C\) and the general solution \(\displaystyle y_{general}(x)=Ay_{2}(x)+By_{1}(x)=\left(A\int e^{-\int(2y_{1}'/y_{1}+a_{1}/a_{2})dx}dx+B+C\right)y_{1}(x)\)

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