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AP EAMCET2005MathematicsDifferential Equations

Observe the following statements I. If d y+2 x y d x=2 e^ -x^2 d x , then y e^ x^2 =2 x+c II. If y e^ -x^2 -2 x=c , then d x= (2 e^ -x^2 -2 x y ) d y which of the following is a correct statement?

Options

  1. ABoth I and II are true
  2. BNeither I nor II is true
  3. CI is false, but II is true
  4. DI is false, II is true

Correct answer

C. I is false, but II is true

Step-by-step solution

I. d y+2 x y d x=2 e^ -x^2 d x d y d x +2 x y=2 e^ -x^2 This is a linear differential equation in y Here, P=2 x, Q=2 e^ -x^2 I.F. =e^ P d x =e^ 2 x d x =e^ x^2 Complete solution is aligned & y e^ x^2 =2 e^ -x^2 e^ x^2 d x+c y e^ x^2 & =2 x+c aligned II. y e^ x^2 -2 x=c On differentiating w.r.t x , we get y e^ x^2 2 x+e^ x^2 d y d x -2=0 aligned & e^ x^2 d y d x & =2-2 x y e^ x^2 & d y d x & =2 e^ -x^2 -2 x y aligned I is true and II is false.

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