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KVPY2015MathematicsDifferential Equations

Let f: R R be a continuous function satisfying f(x)+ ₀^ x t f(t) d t+x²=0 for all x R . Then

Options

  1. A_ x f(x)=2
  2. B_ x - f(x)=-2
  3. Cf(x) has more than one point in common with the x -axis
  4. Df(x) is an odd function

Correct answer

B. _ x - f(x)=-2

Step-by-step solution

f^ (x)+x f(x)+2 x=0 d y d x +x y=-2 x I.F. =e^ x² / 2 y e^ x² / 2 = e^ x² / 2 (-2 x) d x+Ce^ x² / 2 y=-2 e^ x² / 2 +Cy=-2+C e^ i x² / 2 f(0)=0 C=2f(x)=2 [e^ -x² / 2 -1 ]

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