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NDA 2024 Maths · Mathematics · Differential Equations · easy

A solution of the differential equation (d^2y)/(dx^2-4y=0?

Single correct option

Which one of the following is a solution of the differential equation d2ydx2−4y=0\dfrac{d^2y}{dx^2}-4y=0?
  1. A. y=e2xy=e^{2x}
  2. B. y=e4xy=e^{4x}
  3. C. y=sin⁡2xy=\sin2x
  4. D. y=x2y=x^2
Show answer and solution

Answer: (A) y=e2xy=e^{2x}

For y=e2xy=e^{2x}, y′′=4e2xy''=4e^{2x}, so y′′−4y=0y''-4y=0. For y=e4xy=e^{4x}, y′′=16e4x≠4e4xy''=16e^{4x}\ne4e^{4x}; for y=sin⁡2xy=\sin2x, y′′=−4sin⁡2xy''=-4\sin2x so y′′−4y=−8sin⁡2x≠0y''-4y=-8\sin2x\ne0; and for y=x2y=x^2, y′′−4y=2−4x2≠0y''-4y=2-4x^2\ne0. Trap: sin⁡2x\sin2x solves y′′+4y=0y''+4y=0 instead, the sign matters.
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