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COMEDK2025MathematicsDifferential EquationsActual

The solution of the differential equation: x y d y= (x e^x x+e^x ) d x is

Options

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

Correct answer

C. y=e^x x+c

Step-by-step solution

The given differential equation is x y , dy = (x e^x x + e^x) , dx . Rearranging the terms to separate the variables, we have y , dy = x e^x x + e^x x , dx . Simplifying the right side, we get y , dy = ( e^x x + e^x x ) , dx . Integrating both sides, y , dy = (e^x x + e^x x ) , dx . The integral of the right side can be solved by observing that d dx (e^x x) = e^x x + e^x 1 x . Therefore, (e^x x + e^x x ) , dx = e^x x + c . The integral of the left side is y . Thus, y = e^x x + c . Answer: y=e^x x+c

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