Question

On the open interval \((-c, c)\), where \( c \) is a positive real number, \( y(x) \) is an infinitely differentiable solution of the differential equation\[ \frac{dy}{dx} = y^2 - 1 + \cos x, \] with the initial condition \( y(0) = 0 \). Then which one of the following is correct?

• A. \( y(x) \) has a local maximum at the origin.
• B. \( y(x) \) has a local minimum at the origin.
• C. \( y(x) \) is strictly increasing on the open interval \((- \delta, \delta)\) for some positive real number \(\delta\).
• D. \( y(x) \) is strictly decreasing on the open interval \((- \delta, \delta)\) for some positive real number \(\delta\).

Answer 1

The Correct Option is D

The correct option is (D): \( y(x) \) is strictly decreasing on the open interval \((- \delta, \delta)\) for some positive real number \(\delta\).