Differential Equations — Maths STD 12 Science — Question

(b) : Substitute $x+y=z \Rightarrow \frac{d y}{d x}=\frac{d z}{d x}-1$
So, the given equation becomes
$
\begin{aligned}
& \frac{d z}{d x}=\sec ^2 z \Rightarrow d x=\cos ^2 z d z \\
\Rightarrow & d x=\left(\frac{1+\cos 2 z}{2}\right) d z \\
\Rightarrow & x=\frac{1}{2}\left(z+\frac{\sin 2 z}{2}\right)+c_1 (Integrating) \\
\Rightarrow & 2 x=x+y+\frac{\sin 2(x+y)}{2}+c_2 \\
\Rightarrow & 2(x-y)=\sin 2(x+y)-c \\
\Rightarrow & \sin 2(x+y)=2(x-y)+c
\end{aligned}
$