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Manipal MET2015MathematicsDifferentiation

The functions u=e^x x and v=e^x x satisfy the equation

Options

  1. Av d u d x -u d v d x =u^2+v^2
  2. Bd^2 u d x^2 =2 v
  3. Cd^2 v d x^2 =-2 u
  4. DAll of these

Correct answer

D. All of these

Step-by-step solution

We have, u=e^x x and v=e^x x On differentiating both sides w.r.t. x , we get d u d x =e^x x+e^x x=u+v and d v d x =e^x x-e^x x=v-u v d u d x -u d v d x =v(u+v)-u(v-u)=u^2+v^2 Now, d^2 u d x^2 = d u d x + d v d x =u+v+v-u=2 v and d^2 v d x^2 = d v d x - d u d x =(v-u)-(v+u)=-2 u

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