Question

The difference between degree and order of a differential equation that represents the family of curves given by $y^2=a(x+\frac{\sqrt{a}}{2}),a>0$ is

Answer 1

$y^2=a(x+\frac{\sqrt{a}}{2})$
Differentiating with respect to $x,$
$2y{y}^{\prime }=a$
$\Rightarrow y^2=2y{y}^{\prime }(x+\frac{\sqrt{2y{y}^{\prime }}}{2})$
$\Rightarrow y=2y^{\prime }(x+\frac{\sqrt{y{y}^{\prime }}}{\sqrt{2}})$
$\Rightarrow y-2x{y}^{\prime }=\sqrt{2}{y}^{\prime }\sqrt{y{y}^{\prime }}$
$\Rightarrow {(y-2x\frac{dy}{dx})}^2=2y{\left(\frac{dy}{dx}\right)}^3$
Degree $=3$ and Order $=1$
Degree $-$ Order $=3-1=2$