What is the solution of the differential equation ((dy)/(dx))=p...

The given differential equation is $\cos\left(\dfrac{dy}{dx}\right)=p$

$\Rightarrow \dfrac{dy}{dx} = \cos^{-1}(p)$

Integrating both sides with respect to $x$, we get:

$y = x \cos^{-1}(p) + C$

Given $y(0) = q$, substituting $x = 0 and$y = q$:<br><br>$q = 0 + C \Rightarrow C = q$<br><br>Substituting the value of$C$in the equation:<br><br>$y = x \cos^{-1}(p) + q$<br><br>$\Rightarrow y - q = x \cos^{-1}(p)$<br><br>$\Rightarrow \dfrac{y - q}{x} = \cos^{-1}(p)$<br><br>$\Rightarrow \cos\left(\dfrac{y - q}{x}\right) = p$<br><br>Answer:$\cos\left(\dfrac{y-q}{x}\right)=p