Question

The particular solution of the differential equation \( \cos \left( \frac{dy}{dx} \right) = a \), under the conditions \( a \in \mathbb{R} \) and \( y(0) = 2 \) is

• A. \( \cos \left( \frac{x - 2}{y - 2} \right) = a \)
• B. \( \cos^{-1} \left( \frac{y - 2}{x} \right) = a \)
• C. \( \cos \left( \frac{y - 2}{x} \right) = a \)
• D. \( \cos \left( \frac{x - 2}{y + 2} \right) = a \)

Hint:

For separable differential equations, separate variables, integrate, and then apply the initial conditions to find the solution.

Answer 1

The Correct Option is C

Step 1: Solve the differential equation.
Given the equation \( \cos \left( \frac{dy}{dx} \right) = a \), we integrate both sides to get: \[ \frac{dy}{dx} = \cos^{-1} (a) \] Now, integrating with respect to \( x \), we get: \[ y = \frac{y - 2}{x} + C \] Step 2: Apply the initial condition.
Substituting \( y(0) = 2 \) gives the constant \( C \). The solution is \( \cos \left( \frac{y - 2}{x} \right) = a \).
Step 3: Conclusion.
The correct answer is (C) \( \cos \left( \frac{y - 2}{x} \right) = a \).