Question:
The difference between degree and order of differential equation that represents the family of curves given by $y^{2}=a\left(x+\frac{\sqrt{a}}{2}\right), a>0$ is
Solution:
$y^{2}=a\left(x+\frac{\sqrt{a}}{2}\right)$
$2 y y^{\prime}=a$
$y^{2}=2 y y^{\prime}\left(x+\frac{\sqrt{2 y y^{\prime}}}{2}\right)$
$y=2 y^{\prime}\left(x+\frac{\sqrt{y y^{\prime}}}{\sqrt{2}}\right)$
$y-2 x y^{\prime}=\sqrt{2} y^{\prime} \sqrt{y y^{\prime}}$
$\left(y-2 x \frac{d y}{d x}\right)^{2}=2 y\left(\frac{d y}{d x}\right)^{3}$
$D=3 \& O=1$
$D-O=3-1=2$