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QJEEM22Ma3f01: JEE Main 2022 Mathematics - Class 12 JEEM22Ma3f01 Solution: If is the solution of the differential equation such that ,…

If y=y(x)y = y(x) is the solution of the differential equation

2x2dydx−2xy+3y2=02{x^2}{{dy} \over {dx}} - 2xy + 3{y^2} = 0 such that y(e)=e3y(e) = {e \over 3} , then y(1) is equal to :

Multiple Choice Options

  • A 13{1 \over 3}
  • B 23{2 \over 3}
  • C 32{3 \over 2}
  • D 3

Step-by-Step Verified Solution

2x2dydx−2xy+3y2=02{x^2}{{dy} \over {dx}} - 2xy + 3{y^2} = 0

⇒2x(xdy−ydx)+3y3dx=0\Rightarrow 2x(xdy - ydx) + 3{y^3}dx = 0

⇒2(xdy−ydxy2)+3dxx=0\Rightarrow 2\left( {{{xdy - ydx} \over {{y^2}}}} \right) + 3{{dx} \over x} = 0

⇒−2xy+3ln⁡x=C\Rightarrow - {{2x} \over y} + 3\ln x = C

∵\because y(e)=e3⇒−6+3=C⇒C=−3y(e) = {e \over 3} \Rightarrow - 6 + 3 = C \Rightarrow C = - 3

Now, at x=1x = 1 , −2y+0=−3- {2 \over y} + 0 = - 3

y=23y = {2 \over 3}

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