Question

Find the general solution of the differential equation $y=px+p^2$ where $p=\frac{dy}{dx}$

Answer 1

Given the differential equation is $y=px+p^2$.

Now differentiating both sides with respect to $x$ we get,

$p=p+\{x+2p\}\frac{dp}{dx}$ [ Since $p=\frac{dy}{dx}$]

or, $\frac{dp}{dx}=0$ and $x+2p=0$

or, $p=c$......(1)[ Where $c$ is integrating constant] and $x+2p=0$......(2).

Equation (1) will lead us to a general solution and equation (2) will lead us singular solution.

So the general solution is $y=cx+c^2$. [ putting $p=c$ in the given differential equation]