The degree of the differential equation which has a solution $y^2=4a\left(x+a^2\right)$ where a is arbitrary constant 

Differentiating $y^2=4a\left(x+a^2\right)$

We have,  $2y\frac{dy}{dx}=4a\Rightarrow a=\frac{y{y}^{\mathrm{\prime }}}{2}$

Putting in $\left(1\right)$ 

$\begin{array}{l}{y}^2=4\frac{y{y}^{\mathrm{\prime }}}{2}\left(x+\frac{1}{4}{y}^2{y}^2\right)\\ =2xy{y}^{\mathrm{\prime }}+y^3{y}^{\mathrm{\prime }3}\end{array}$

So the degree is 3