$\cos \left( {\cfrac{{dy}}{{dx}}} \right) = a(a \in R);y = 1$ when $x = 0$

.: We have, $\cos \left( {\cfrac{{dy}}{{dx}}} \right) = a$

$\Rightarrow \cfrac{{dy}}{{dx}} = {\cos ^{ - 1}}a \Rightarrow dy = {\cos ^{ - 1}}adx$ …(1)

Integrating (1) both sides,

we get
$\int d y = {\cos ^{ - 1}}a\int {dx} \Rightarrow y = x{\cos ^{ - 1}}a + C$

When $x = 0,y = 1 \Rightarrow 1 = C$

Thus, particular solution is $y = x{\cos ^{ - 1}}a + 1$

$\Rightarrow (y - 1) = x{\cos ^{ - 1}}a \Rightarrow {\cos ^{ - 1}}a = \left( {\cfrac{{y - 1}}{x}} \right)$

$\Rightarrow a = \cos \left( {\cfrac{{y - 1}}{x}} \right)$