Question

The particular solution of the differential equation $\cos \left(\frac{dy}{dx}\right)=a,\left(a\in R\right)$, satisfying the condition n, $y=2$ when x = 0 is:

• A. $\cos \left(\frac{y+2}{x}\right)=a$
• B. $\cos \left(\frac{x+2}{y}\right)=a$
• C. $\cos \left(\frac{y-2}{x}\right)=a$
  
• D. $\cos \left(\frac{x-2}{y}\right)=a$

Answer 1

The correct option is C $\cos \left(\frac{y-2}{x}\right)=a$

Given differential equation , $\cos \left(\frac{dy}{dx}\right)=a$

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

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

Integrating both sides

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

Now when $y=2x=0$, we get

$2=c$

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

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