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AP EAMCET201726 Apr 2017Morning ShiftMathematicsDifferential EquationsActual

The solution of the equation (x-4 y^3 ) d y d x -y=0,(y>0) is

Options

  1. Ax=y^3+c y
  2. Bx+2 y^3=c y
  3. Cy=x^3+c x
  4. Dy+2 x^3=c x

Correct answer

B. x+2 y^3=c y

Step-by-step solution

Here, the differential equation aligned & (x-4 y^3 ) d y d x -y=0,(y>0) & (x-4 y^3 ) d y d x =y d x d y = x-4 y^3 y & d x d y = x y -4 y^2 d x d y - x y =-4 y^2 & P=- 1 y Q=-4 y^2 aligned Integrating factor, I.F. =e^ P d y =e^ - 1 y d y =e^ - y =e^ y-1 = 1 y Now, the solution is given by aligned & x ( I.F. )= ( I.F. ) Q d y+C & x y = 1 y (-4 y^2 ) d y+C & x y = -4 y d y+C & x y =-4 y^2 2 +C aligned aligned & x=-2 y^3+C y & x+2 y^3=C y aligned This is the required solution.

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