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AP EAMCET202124 Aug 2021Evening ShiftMathematicsDifferential EquationsActual

The equation of a curve passing through the point (0,1) , given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x -coordinate and the product of x and y coordinates at that point, is

Options

  1. Ay=1-2 e^ ( x^2 2 )
  2. By=-1+2 e^ ( x^2 2 )
  3. Cy=-1-2 e^ ( x^2 2 )
  4. Dy=1+2 e^ ( x^2 2 )

Correct answer

B. y=-1+2 e^ ( x^2 2 )

Step-by-step solution

Given, d y d x =x+x y d y d x =x(1+y) d y 1+y =x d x On integrating, we get (y+1)= x^2 2 +C y+1=A e^ x^2 / 2 y=A e^ x^2 / 2 -1 Since it passes through (0,1) . 1=A-1 A=2 Equation of curve is y=2 e^ x^2 / 2 -1

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