Question

Solve: $\frac{dy}{dx}=\cos (x+y)$

Answer 1

Let $x+y=2v$

$\Rightarrow 1+\frac{dy}{dx}=2\frac{dv}{dx}$

So given differential equation becomes,

$2\frac{dv}{dx}-1=\cos 2v$

$\Rightarrow 2\frac{dv}{dx}=1+\cos 2v=2{\cos }^{2}v$

$\Rightarrow \frac{dv}{dx}={\cos }^{2}v$

$\Rightarrow {\sec }^{2}vdv=dx$

Now integrate both sides,

$\tan v=x+C$

$\Rightarrow \tan \frac{x+y}{2}=x+C$