Answer
\[x^2+2y=K\]
Work Step by Step
$\large{y=\ln (cx)}$ ____(1)
Differentiate (1) with respect to $x$
$\Large\frac{dy}{dx}=\Large\frac{c}{cx}=\Large\frac{1}{x}$ ____(2)
Replace $\Large\frac{dy}{dx}$ by $-\Large\frac{dx}{dy}$ in (2) [For orthogonal trajectories]
\[-\frac{dx}{dy}=\frac{1}{x}\]
Separating variables,
$$-x dx=dy$$
Integrating,
$$C_{1}-\int x dx=\int dy$$
$$C_{1}-\frac{x^2}{2}=y$$
$$x^2+2y=K$$
Where $K=2C_{1}$
Hence orthogonal trajectories of (1) is $x^2+2y=K$
