Discussion Forum

The general solution of differential equation $\frac{dx}{dy}=cos(x+y)$  is

Answer:

Explanation:

We have ,   $\frac{dx}{dy}=cos(x+y)$

 Put , x+y=v

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

$\Rightarrow$     $\frac{dx}{dy}=\frac{dv}{dy}-1$

$\therefore$   Equation  becomes

 $\frac{dv}{dy}-1= \cos v$

$\Rightarrow$  $\frac{dv}{dy}=1+\cos v$

$\Rightarrow$   $\frac{dv}{1+\cos v}+dy$

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

 On integrating , we get

$\frac{1}{2}\int \sec^{2}\frac{v}{2}dv=\int dy$

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

 $\Rightarrow$    $\tan\left(\frac{x+y}{2}\right)=y+C$     [$\therefore$   v=x+y]