Differential Equations — Maths STD 12 Science — Question

$x+y \frac{d y}{d x}=\sec \left(x^2+y^2\right)$
Put $x^2+y^2=v \Rightarrow 2 x+2 y \frac{d y}{d x}=\frac{d v}{d x}$
$x+y \frac{d y}{d x}=\frac{1}{2} \frac{d v}{d x}$
$\therefore$ Given equation becomes,
$\frac{1}{2} \frac{d v}{d x}=\sec v$
$\therefore \cos v d v=2 d x$
$\therefore$ Integrating both sides,
$\int \cos v d v=2 \int d x$
$\therefore \sin v=2 x+c$
Substituting the value of $v$,
$\sin \left(x^2+y^2\right)=2 x+c$