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How was this $y\left(y’’+\frac{1-p}vy’\right)-(y+1-p)y’^2=0$ power series recurrence derived?

In György Steinbrecher’s and William Shaw’s Quantile mechanics $(47)$ to $(51)$, it can be found that: $$\begin{aligned}y\left(y’’+\frac{1-p}vy’\right)-(y+1-p)y’^2=0,y(0)=0,y (0)=1\iff (1-p)y’^2=\...
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