Which of the following functions is a solution of the differential equation \(\displaystyle\frac{d^2y}{dx^2} + 9y = 0\)?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(y = 2\sin 3x - \cos 3x\)
A function is a solution when it satisfies the equation identically. For \(y = 2\sin 3x - \cos 3x\):
\( \displaystyle \frac{dy}{dx} = 6\cos 3x + 3\sin 3x \)
\( \displaystyle \frac{d^2y}{dx^2} = -18\sin 3x + 9\cos 3x \)
\( \displaystyle = -9(2\sin 3x - \cos 3x) = -9y \)
So \(\frac{d^2y}{dx^2} + 9y = 0\).
Why the other options are incorrect
Option (1) \(y = e^{3x}\) — Incorrect. \(\frac{d^2y}{dx^2} = 9e^{3x}\), so the left side is \(18e^{3x} \ne 0\).
Option (2) \(y = \sin 9x\) — Incorrect. \(\frac{d^2y}{dx^2} = -81\sin 9x\), so the left side is \(-72\sin 9x\), not zero.
Option (3) \(y = x\sin 3x\) — Incorrect. \(\frac{d^2y}{dx^2} = 6\cos 3x - 9x\sin 3x\), so the left side is \(6\cos 3x\), not zero.