Question

The solution of the equation $\frac{dy}{dx}=cos(x-y)$ is

• A. $y+cot\left(\frac{x-y}{2}\right)=C$
• B. $x+cot\left(\frac{x-y}{2}\right)=C$
• C. $x+tan\left(\frac{x-y}{2}\right)=C$
• D. None of these

Answer 1

The correct option is B $x+cot\left(\frac{x-y}{2}\right)=C$

Putting $u=x-y$, we get $\frac{du}{dx}=1-\frac{dy}{dx}$. The given equation can be written as $1-\frac{du}{dx}=cosu$
$\Rightarrow (x-cosu)=\frac{du}{dx}\Rightarrow \int \frac{du}{1-cosu}=\int dx+c\Rightarrow \frac{1}{2}\int cose{c}^2\left(\frac{u}{2}\right)du=\int dx+C\Rightarrow x+cot\left(\frac{u}{2}\right)=c\Rightarrow x+cot\left(\frac{x-y}{2}\right)=C$