Verified Solution • 2025–26 Session moderate

QJEEM21Ma7383: JEE Main 2021 Mathematics - Class 12 JEEM21Ma7383 Solution: Which of the following is true for y(x) that satisfies the…

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

dydx{{dy} \over {dx}} = xy −- 1 + x −- y; y(0) = 0 :

Multiple Choice Options

  • A y(1) = 1
  • B y(1) = e −- 12{1 \over 2} −- 1
  • C y(1) = e 12{1 \over 2} −- e −- 12{1 \over 2}
  • D y(1) = e 12{1 \over 2} −- 1

Step-by-Step Verified Solution

dydx=(x−1)y+(x−1){{dy} \over {dx}} = (x - 1)y + (x - 1)

dydx=(x−1)(y+1){{dy} \over {dx}} = (x - 1)(y + 1)

dyy+1=(x−1)dx{{dy} \over {y + 1}} = (x - 1)dx

Integrating both sides, we get

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

x=0,y=0x = 0,y = 0

⇒c=0\Rightarrow c = 0

∴\therefore ln⁡(y+1)=x22−x\ln (y + 1) = {{{x^2}} \over 2} - x

putting x=1,ln⁡(y+1)=12−1=−12x = 1,\ln (y + 1) = {1 \over 2} - 1 = - {1 \over 2}

y+1=e−12y + 1 = {e^{ - {1 \over 2}}}

y=e−12−1y = {e^{ - {1 \over 2}}} - 1

∴\therefore y(1)=e−12−1y(1) = {e^{ - {1 \over 2}}} - 1

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