Question:
$\cos \left(\frac{d y}{d x}\right)=a(a \in R) ; y=1$ when $x=0$
Solution:
$\cos \left(\frac{d y}{d x}\right)=a$
$\Rightarrow \frac{d y}{d x}=\cos ^{-1} a$
$\Rightarrow d y=\cos ^{-1} a d x$
Integrating both sides, we get:
$\int d y=\cos ^{-1} a \int d x$
$\Rightarrow y=\cos ^{-1} a \cdot x+\mathrm{C}$
$\Rightarrow y=x \cos ^{-1} a+\mathrm{C}$ ...(1)
Now, $y=1$ when $x=0$
$\Rightarrow 1=0 \cdot \cos ^{-1} a+C$
$\Rightarrow C=1$
Substituting C = 1 in equation (1), we get:
$y=x \cos ^{-1} a+1$
$\Rightarrow \frac{y-1}{x}=\cos ^{-1} a$
$\Rightarrow \cos \left(\frac{y-1}{x}\right)=a$
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