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

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

$$a_{2}(x)\frac{d^{2}y}{dx^{2}}+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y(x)=0$$

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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