How to find the singular solution of $8ap^3=27y$ where $p=\frac{dy}{dx}$?
My effort:
$8ap^3=27y$
Differentiating both sides we get
$8a(3p^2)\frac{dp}{dx}=27p \implies 8ap^2\frac{dp}{dx}=9p\implies p(8ap\frac{dp}{dx}-9)=0\implies $ either $p=0$ or $8ap\frac{dp}{dx}=9$.
If $p=0$ then $y$ is constant.
Can someone please help me to find the singular solution of this differential equation?
A singular solution is a solution that cannot be obtained from the general integral (or can be obtained as a limit solution).
For equations in which the variables separate, as in $$ y' = g(x)h(y) $$ there is a constant singular solution $y(t) = y_0$ for each value $y_0$ such that $h(y_0)=0$.
In this case, given that the equation can be written $$ y' = \sqrt[3]{\frac{27}{8a}y} $$ there is a singular solution $y(t) = 0$.