The differential equation of the family of curves,

Since, $x^{2}=4 b(y+b)$

$x^{2}=4 b y+4 b^{2}$

$2 x=4 b y^{\prime}$

$\Rightarrow \quad b=\frac{x}{2 y^{\prime}}$

So, differential equation is

$x^{2}=\frac{2 x}{y^{\prime}} \cdot y+\left(\frac{x}{y^{\prime}}\right)^{2}$

$x\left(y^{\prime}\right)^{2}=2 y y^{\prime}+x$