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

Answer 1

Answer: 2

$y^2=a\left(x+\frac{\sqrt{a}}{2}\right)=ax+\frac{a^{3/2}}{2}\dots (1)$
$\Rightarrow 2y{y}^{\prime }=a$
put in equation (1)
$y^2=\left(2y{y}^{\prime }\right)x+\frac{{\left(2y{y}^{\prime }\right)}^{3/2}}{2}$
$\left(y^2-2xy{y}^{\prime }\right)=\frac{{\left(2y{y}^{\prime }\right)}^{3/2}}{2}$
squaring
${\left(y^2-2xy{y}^{\prime }\right)}^2=\frac{y^3{\left(y^{\prime }\right)}^3}{2}$
$\therefore$ order $=1$
degree $=3$
Degree $-$ order $=3-1=2$