Question

Form the differential equation representing the family of curves given by $(xa{)}^2+2y^2=a^2$, where $a$ is an arbitrary constant

Answer 1

Given, $(xa{)}^2+2y^2=a^2$

Differentiating it once: $2x{a}^2+4y{y}^{{}^{\prime }}=0$

$a^2=\frac{-2y{y}^{{}^{\prime }}}{x}$

Putting this in main equation:

$-2xy{y}^{{}^{\prime }}+2y^2=\frac{-2y{y}^{{}^{\prime }}}{x}$