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AP EAMCET2002MathematicsDifferential Equations

y+x^2= d y d x has the solution

Options

  1. Ay+x^2+2 x+2=c e^x
  2. By+x+2 x^2+2=c e^x
  3. Cy^2+x+x^2+2=c e^ 2 x
  4. Dy+x+x^2+2=c e^ 2 x

Correct answer

A. y+x^2+2 x+2=c e^x

Step-by-step solution

We have, aligned y+x^2 & = d y d x d y d x -y & =x^2 aligned It is a linear differential equation. On comparing with d y d x +P y=Q , we get aligned & P=-1, Q=x^2 & IF =e^ P d x =e^ -1 d x =e^ -x aligned Required solution is aligned & & y e^ -x & = x^2 e^ -x d x+c & & y e^ -x & =-x^2 e^ -x +2 x e^ -x d x & & y e^ -x & =-x^2 e^ -x -2 x e^ -x +2 e^ -x d x & & y e^ -x & =-x^2 e^ -x -2 x e^ -x -2 e^ -x +c & & y & =- (x^2+2 x+2 )+c e^x & & y & +x^2+2 x+2=c e^x aligned

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