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JEE Main 202117 MarchMorning Shift
Exam
JEE Main
Year
2021
Date
17 March
Shift
Morning Shift
Subject
Mathematics
Chapter
Differential Equations

JEE Main 2021 · 17 March · Morning Shift · Mathematics · Differential Equations

Which of the following is true for y(x) that satisfies the...

Which of the following is true for y(x)y\left(x\right) that satisfies the differential equation

dydx=xy−1+x−y;y(0)=0\dfrac{dy}{dx} = xy - 1 + x - y; y\left(0\right) = 0

Options

  1. Ay(1)=e−12−1y\left(1\right) = {e}^{- \dfrac{1}{2}} - 1
  2. By(1)=e12−e−12y\left(1\right) = e\dfrac{1}{2} - {e}^{- \dfrac{1}{2}}
  3. Cy(1)=1y\left(1\right) = 1
  4. Dy(1)=e12−1y\left(1\right) = e\dfrac{1}{2} - 1

Correct answer

A. y(1)=e−12−1y\left(1\right) = {e}^{- \dfrac{1}{2}} - 1

Step-by-step solution

Given dydx=(1+y)(x−1)\dfrac{dy}{dx} = \left(1 + y\right)\left(x - 1\right)

⇒dy(y+1)=(x−1)dx⇒ \dfrac{dy}{( y + 1 )} = \left(x - 1\right)dx

Integrate both the sides we get,

ln⁡(y+1)=x22−x+c\ln \left(y + 1\right) = \dfrac{{x}^{2}}{2} - x + c

Given y(0)=0⇒c=0y\left(0\right) = 0 ⇒ c = 0

∴y=e(x22−x)−1∴ y = {e}^{\left(\dfrac{{x}^{2}}{2} - x\right)} - 1

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