how to reduce the equation x^2(y-px)=p^2y to clairaut form and find it’s general solution

Question : Reduce the equation x^2(y-px)=p^2y to clairaut form and find it’s general solution Solution : To Reduce the equation \[x^2\left(y-px\right)=p^2y\] \[y-px=\frac{y}{x^2}p^2\] \[x^2-\frac{x^3}{y}p=p^2 \] (1) Let \[X=x^2,\ Y=y^2\] Then \[\frac{dX}{dx}=2x,\frac{dY}{dx}=2y\frac{dy}{dx}\] \[\frac{dy}{dx}=\frac{1}{2y}\frac{dY}{dx}\] \[\frac{dy}{dx}=\frac{1}{2y}\frac{dY}{dX}\frac{dX}{dx}=\frac{1}{2y}\frac{dY}{dX}2x=\frac{x}{y}\frac{dY}{dX}\] \[p=\frac{x}{y}P\] Substituting in eqn (1) \[X-\frac{X^2}{Y}P=\frac{X}{Y}P^2\] \[X=\frac{1}{Y}\left(X^2P+XP^2\right)\] \[Y=XP+P^2\]…

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