Question

For each of different example, find a particular solution satisfying the given example.
$\cos \left(\frac{dy}{dx}\right)=a;y=1,x=0$

Answer 1

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

$\frac{dy}{dx}={\cos }^{-1}a$

$dy={\cos }^{-1}a.dx$

$\int dy=\int {\cos }^{-1}a.dx$

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

Put $x=0,y=1$

$1=0{\cos }^{-1}a+c$

$1=c$

put $c$ in (1)

$y=x{\cos }^{-1}a+1$

$y-1=x{\cos }^{-1}a$

$\frac{y-1}{x}={\cos }^{-1}a$

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