Question

Find the general solution of the DE
y² lny = x py+p².
Does the equation has any singular solution? If yes, obtain it.

Answer 1

Let's rewrite the equation in form

$\ln y=x\frac{p}{y}+\frac{p^2}{y^2}$

Denote $Y=\ln y$ hence $P=\frac{dY}{dx}=\frac{p}{y}$

Terefore

$Y=xP+P^2$

which is in the Clairaut's form

The solution can be obtained by just replacing P by constant c.

$Y=cx+c^2$

or

$\ln y=cx+c^2$

$y=\exp(cx+c^2)$

To find a singular solution, let's solve the equation

$(P^2)'+x=0$

$2P=-x$

$P=-\frac{x}{{2}}$

Therefore

$Y=-\frac{x^2}{2}+(-\frac{x}{2})^2=-\frac{1}{4}x^2$

will be the singular solution of this equation