The particular solution of the differential equation $\cos \left(\frac{d y}{d x}\right)=a,(a \in R) ; y=2$ at x = 0 is given by

$\cos \left(\frac{y-2}{x}\right)=a$

The correct answer is Option (1) → $\cos \left(\frac{y-2}{x}\right)=a$

$\cos\frac{dy}{dx}=a⇒\int\, dy=\cos^{-1}a\int x\,dx$

so $y+c=(\cos^{-1}a)(x)$

at $x=0$, $y=2$

so $c=-2$

$⇒\frac{y-2}{x}=\cos^{-1}a⇒\cos\left(\frac{y-2}{x}\right)=a$