Determine the general and singular solutions of the equation
$$ y= xp + \frac{ap}{(1+p^2)^{1/2}} $$
where $a$ is a constant and $p=\frac{dy}{dx}$.
Since the given differential equation is in Clairaut's form, the general solution can be obtained by replacing $p$ with some $c$ (constant).
But for singular solutions, I couldn't find them as the equations are complex.
Please help me with this.
Hint: Derivation respect to $x$ of $y= xp + \dfrac{ap}{(1+p^2)^{1/2}}$ gives $p=y'=p+xp'+\cdots$ which simplify to $$xp'+\dfrac{ap'}{\sqrt{1+p^2}^3}=0$$ now we can drop $p'$ from this, $\sqrt{1+p^2}=\sqrt[3]{-\dfrac{a}{x}}$ and from the main equation and find the result.