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NDA 1 2024 Maths, question 72

Topic: Differential equations

Q72

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

  1. (a) d2ydx2+2y=0\frac{d^{2}y}{dx^{2}}+2y=0
  2. (b) d2ydx2+2dydx+2y=0\frac{d^{2}y}{dx^{2}}+2\frac{dy}{dx}+2y=0
  3. (c) d2ydx2−2dydx+2y=0\frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+2y=0Correct answer
  4. (d) d2ydx2+y=0\frac{d^{2}y}{dx^{2}}+y=0

Solution.

Answer: (c) d2ydx2−2dydx+2y=0\frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+2y=0

  1. For y = e^x(a cos x + b sin x), differentiation twice gives y′ = y + e^x(−a sin x + b cos x), then y″ = 2y′ − 2y.
  2. Thus y″ − 2y′ + 2y = 0.