Question

For the following differential equation, find a particular solution satisfying the given condition:- \[\cos\left( \frac{dy}{dx} \right) = a, y = 1\text{ when }x = 0\]

Answer 1

We have,

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

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

\[ \Rightarrow dy = \cos^{- 1} a dx\]

Integrating both sides, we get

\[\int dy = \int \cos^{- 1} a dx\]

\[ \Rightarrow y = x \cos^{- 1} a + C\]

Now,

When `x = 0, y = 1`

\[ \therefore 1 = 0 + C\]

\[ \Rightarrow C = 1\]

Putting the value of `C` in (1), we get

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

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