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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:

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

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

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