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JEE Advanced2022MathematicsDifferential EquationsActual

For x ∈ ℝ , let the function y x be the solution of the differential equation d y d x + 12 y = cos π 12 x , y 0 = 0 . Then, which of the following statements is/are TRUE?

Options

  1. Ay x is an increasing function
  2. By x is a decreasing function
  3. CThere exists a real number β such that the line y = β intersects the curve y = y x at infinitely man
  4. Dy x is a periodic function

Correct answer

C. There exists a real number β such that the line y = β intersects the curve y = y x at infinitely man

Step-by-step solution

Given, d y d x + 12 y = cos π x 12 I . F = e 12 x Now solution of the differential equation is given by, ⇒ y · e 12 x = ∫ e 12 x · cos π x 12 d x + C ⇒ y · e 12 x = e 12 x 12 2 + π 12 2 12 cos π x 12 + π 12 sin π x 12 + C Now given curve is passing through 0 , 0 So, y 0 = 0 ⇒ C = - 12 12 2 + π 12 2 So y = 1 λ 12 cos π x 12 + π 12 sin π x 12 ⏟ f 1 x - 12 e - 12 x Where λ = 1 12 2 + π 12 2 Now on differentiat

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