Question

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

• A. $\tan \left(\frac{x+y}{2}\right)=y+c$
• B. $\tan \left(\frac{x+y}{2}\right)=x+c$
• C. $\cot \left(\frac{x+y}{2}\right)=y+c$
• D. $\cot \left(\frac{x+y}{2}\right)=x+c$

Answer 1

The correct option is A $\tan \left(\frac{x+y}{2}\right)=y+c$

Let $x+y=v⟹\frac{dx}{dy}+1=\frac{dv}{dy}$

$⟹\frac{dv}{dy}=1+\cos v=2{\cos }^2\frac{v}{2}$

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

$⟹{\sec }^2\frac{v}{2}dv=2dy$

Integrating both sides:-

$2\tan \frac{v}{2}=2y+k$

$\tan \left(\frac{x+y}{2}\right)=y+c$

Hence, answer is option-(A).