Question

Find the particular solution of $\cos \left(\frac{dy}{dx}\right)=\frac{1}{3},y\left(0\right)=2$.

Answer 1

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

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

$\Rightarrow dy={\cos }^{-1}adx$

Integrating both sides, we get

$\int dy=\int {\cos }^{-1}adx$

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

Putting $x=0$ and $y=2$

$\Rightarrow 2=0{\cos }^{-1}a+c$

$\therefore c=2$

Putting the value of $c$ in $\left(1\right)$ we get

$y=x{\cos }^{-1}a+2$

$\Rightarrow y-2=x{\cos }^{-1}a$

$\Rightarrow {\cos }^{-1}a=\frac{y-2}{a}$

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