Question

Find particular solution of differential equation $cos\left(\frac{dy}{dx}\right)=\frac{1}{5},y\left(0\right)=2$

Answer 1

$\cos \left(\frac{dy}{dx}\right)=\frac{1}{5}$

$\frac{dy}{dx}={\cos }^{-1}\left(\frac{1}{5}\right)$

$\Rightarrow dy={\cos }^{-1}\left(\frac{1}{5}\right)dx$

Integrating both sides, we have

$\int dy=\int {\cos }^{-1}\left(\frac{1}{5}\right)dx$

$y=x{\cos }^{-1}\left(\frac{1}{5}\right)+C$

Given that:- $y\left(0\right)=2$

$\therefore 2=0+C$

$\Rightarrow C=2$

Therefore,

$y=x{\cos }^{-1}\left(\frac{1}{5}\right)+2$

Hence the required solution is $y=x{\cos }^{-1}\left(\frac{1}{5}\right)+2$.