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