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BITSAT2018MathematicsDifferential EquationsActual

The equation of the curve whose slope at any point is equal to y + 2 x and which passes through the origin is

Options

  1. Ay = 2 x - 1
  2. By = 2 e x - x - 1
  3. Cy = 2 e x - 1
  4. Dy = 2 e x x - 1

Correct answer

B. y = 2 e x - x - 1

Step-by-step solution

We have, d y d x = y + 2 x ⇒ d y d x - y = 2 x Now, IF = e - ∫ 1 d x = e - x y · e - x = ∫ 2 x e - x d x + k ,   k be the constant of integration = 2 x ∫ e - x d x - ∫ 1 · - e - x d x + k [using integration by parts] ⇒ y · e - x = - 2 x e - x - 2 e - x + k       . . . i As curve (i) passes through 0 , 0 ∴   0 = 0 - 2 + k ⇒ k = 2 Thus, the curve is y e - x = - 2 x e - x - 2 e - x + 2 ∴   y = 2 e x - x - 1 Hence, option

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