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BITSAT2018MathematicsDifferential EquationsActual

Solution of the differential equation d y d x = sin x + y + cos x + y is equal to

Options

  1. Alog 2 + sec x + y 2 = x + C
  2. Blog 1 + tan x + y = x + C
  3. Clog 1 + tan x + y 2 = y + C
  4. Dlog 1 + tan x + y 2 = x + C

Correct answer

D. log 1 + tan x + y 2 = x + C

Step-by-step solution

We have, d y d x = sin x + y + cos x + y Put x + y = v ⇒ 1 + d y d x = d v d x d v d x - 1 = sin v + cos v ⇒ d v d x = sin v + cos v + 1 ⇒ d v sin v + cos v + 1 = d x ⇒ ∫ 1 sin v + cos v + 1 d v = ∫ d x ⇒ ∫ 1 2 tan v 2 1 + tan 2 v 2 + 1 - tan 2 v 2 1 + tan 2 v 2 + 1 d v = ∫ d x ⇒ ∫ 1 + tan 2 v 2 2 tan v 2 + 1 - tan 2 v 2 + 1 + tan 2 v 2 d v = x + C ⇒ 1 2 ∫ sec 2 v 2 tan v 2 + 1 d v = x + C ⇒ log tan v 2 + 1 = x + C ⇒ log tan x

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