Question

Reduce the equation x²(y-px)=yp² to clairaut´s form and hence find its complete solution.

Answer 1

Let's put $X=x^2;\quad Y=y^2$ then $p=\frac{x}{y}\frac{dY}{dX}$ . Let's denote $P=\frac{dY}{dX}$

Then the equation can be rewritten in form

$X(y-x^2\frac{P}{y})=y\frac{x^2}{y^2}P^2$

By multiplying both sides with y, we assume

$X(y^2-x^2P)=x^2P^2$

Or

$X(Y-XP)=XP^2$

Therefore

$Y=XP+P^2$

which is now in Clairaut’s form

The solution got by just replacing P by constant c.

Hence

$Y=cX+c^2$

or

$y^2=cx^2+c^2$