Thank you for reporting, we will resolve it shortly
Q.
Orthogonal trajectories of family of the curve $x^{2/3}+y^{2/3}=a^{2/3}$, where a is any arbitrary constant, is
Differential Equations
Solution:
$x^{2/3}+y^{2/3}=a^{2/3}$
or $\frac{2}{3} x^{-1/3}+\frac{2}{3} y^{-1/3} \frac{dy}{dx}=0$
or $\frac{dy}{dx}=-\frac{x^{-1/3}}{y^{-1/3}} (1)$
Replacing $\frac{dy}{dx} \left(\frac{\pi}{2}-\theta\right)$ by $-\frac{dx}{dy}$, we get
$\frac{dx}{dy}=\frac{x^{-1/3}}{y^{-1/3}}$
or $\int x^{1 /3} dx=\int y^{1 /3}dy$
or $x^{4/ 3}-y^{4/ 3}=c$