Question:
The difference between degree and order of a differential equation that represents the family of curves given by $y^{2}=a\left(x+\frac{\sqrt{a}}{2}\right), a>0$ is
Solution:
$y^{2}=a\left(x+\frac{\sqrt{a}}{2}\right)=a x+\frac{a^{3 / 2}}{2}$
$\Rightarrow 2 \mathrm{yy}^{\prime}=\mathrm{a}$
put in equation (1)
$y^{2}=\left(2 y y^{\prime}\right) x+\frac{\left(2 y y^{\prime}\right)^{3 / 2}}{2}$
$\left(\mathrm{y}^{2}-2 \mathrm{xy} \mathrm{y}^{\prime}\right)=\frac{\left(2 \mathrm{yy}^{\prime}\right)^{3 / 2}}{2}$
squaring
$\left(y^{2}-2 x y y^{\prime}\right)^{2}=\frac{y^{3}\left(y^{\prime}\right)^{3}}{2}$
$\therefore$ order $=1$
degree $=3$