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

Answer: (D)

$y(x)$ is strictly decreasing on the open interval $(- \delta, \delta)$ for some positive real number $\delta$.

Answer 2

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