The differential equation, representing the curve y=e^{x}(a \cos x+b \sin x) where a and b are arbitrary constants, is

  1. A. \frac{d^{2}y}{dx^{2}}+2y=0
  2. B. \frac{d^{2}y}{dx^{2}}+2\frac{dy}{dx}+2y=0
  3. C. \frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+2y=0 ✓
  4. D. \frac{d^{2}y}{dx^{2}}+y=0

Correct Answer: C. \frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+2y=0

Explanation

Find first derivative y' = e^x(a\cos x + b\sin x) + e^x(-a\sin x + b\cos x) = y + e^x(-a\sin x + b\cos x). Find second derivative y'' = y' + e^x(-a\sin x + b\cos x) + e^x(-a\cos x - b\sin x) = y' + (y'-y) - y = 2y' - 2y. Rearranging gives y'' - 2y' + 2y = 0.

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