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AP EAMCET201825 Apr 2018Morning ShiftMathematicsDifferential EquationsActual

If - π 4 < x < π 4 , then the general solution of the differential equation cos 2 x · d y d x - ( tan 2 x ) y = cos 4 x is

Options

  1. Ay = 1 2 tan 2 x + c 1 - tan 2 x
  2. By = 1 2 cos 2 x + c 1 - tan 2 x
  3. Cy = 1 2 sin 2 x + c 1 - tan 2 x
  4. Dy = 1 2 sin x + c 1 - tan 2 x

Correct answer

C. y = 1 2 sin 2 x + c 1 - tan 2 x

Step-by-step solution

Given, cos 2 x · d y d x - ( tan 2 x ) y = cos 4 x ⇒ d y d x - tan 2 x cos 2 x y = cos 4 x cos 2 x ⇒ d y d x - tan 2 x cos 2 x y = cos 2 x This is the linear differential equation of the form d y d x + P y = Q . Therefore, we get P = - tan 2 x cos 2 x and Q = cos 2 x I.F. = e ∫ P d x Now, ∫ P d x = - ∫ tan 2 x cos 2 x d x = - ∫ 2 tan x 1 - tan 2 x sec 2 x d x Let tan x = t ⇒ sec 2 x d x = d t ∫ P d x = ∫ - 2 t 1 - t 2 d t = log e 1 - t 2 e ∫ P d x = e

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