Question

The general solution to the ordinary differential equation \( \dfrac{d^2y}{dx^2} = \cos 2x \) is

• A. \( y = -\dfrac{\cos 2x}{4} + C_1 x + C_2 \), where \( C_1 \) and \( C_2 \) are constants
• B. \( y = \dfrac{\sin 2x}{2} + C_1 \), where \( C_1 \) is a constant
• C. \( y = \dfrac{\cos 2x}{4} + C_1 x + C_2 \), where \( C_1 \) and \( C_2 \) are constants
• D. \( y = -\dfrac{\sin 2x}{2} + C_1 \), where \( C_1 \) is a constant

Hint:

Integrate both sides of the equation twice with respect to x, adding a constant each time.

Answer 1

The Correct Option is A

This is a direct double integration problem, and checking each option by differentiating twice is a fast way to confirm the answer.

1. Option (A): differentiating $y = -\cos(2x)/4 + C_1 x + C_2$ twice gives back $\cos 2x$, so this option satisfies the equation and has the correct two constants for a second-order ODE.
2. Option (B): differentiating $y = \sin(2x)/2 + C_1$ twice gives $-2\sin 2x$, not $\cos 2x$, and it also carries only one constant when a second-order equation needs two.
3. Option (C): differentiating $y = \cos(2x)/4 + C_1 x + C_2$ twice gives $-\cos 2x$, the wrong sign.
4. Option (D): differentiating $y = -\sin(2x)/2 + C_1$ twice gives $2\sin 2x$, not $\cos 2x$, and again only one constant appears.

Only option (A) reproduces $\cos 2x$ after two derivatives and carries the two independent constants a second-order ODE needs, so it is the general solution.