Chapter 1 - First-Order Differential Equations - 1.1 Differential Equations Everywhere - Problems - Page 11: 14

\[5y^2+x^2=K\]

Given that, $y=cx^5$ _____(1) Differentiating both side with respect to $x$ \[\frac{dy}{dx}=5cx^4\] From (1) $\frac{dy}{dx}=5(\frac{y}{x^5})x^4=\frac{5y}{x}$ ____(2) Replace $\frac{dy}{dx}$ by $\frac{-dx}{dy}$ in (2) [ For orthogonal trajectories] $\frac{-dx}{dy}=\frac{5y}{x}$ $-xdx = 5ydy$ Integrating, $-\int xdx= \int 5ydy$ $k-\frac{x^2}{2}=\frac{5y^2}{2}$ $5y^2+x^2=K$, where $K = 2k$ Hence family of orthogonal trajectries to (1) is $5y^2+x^2=K$.