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MHT CET202520 Apr 2025Morning ShiftMathematicsDifferential EquationsActual

The differential equation whose solution represents the family x^2 y=4 e^x+c , where c is an arbitrary constant, is

Options

  1. Ax dy d x +x y =0
  2. Bx^2 dy d x +(2 xy-4e^x)=0
  3. Cx dy d x +(x-2) y =0
  4. Dx dy d x +(2-x) y =0

Correct answer

B. x^2 dy d x +(2 xy-4e^x)=0

Step-by-step solution

Given the solution x^2 y=4 e^x+c , we eliminate the arbitrary constant c through differentiation. Differentiating both sides with respect to x using the product rule for the left-hand side: d d x (x^2 y) = x^2 d y d x + 2x y The derivative of the right-hand side yields: d d x (4e^x + c) = 4e^x Equating both derivatives: x^2 d y d x + 2x y = 4e^x Rewriting in standard differential equation form: x^2 d y d x + (2x y - 4e^x) = 0 This corresponds to option B . The differential equation is B

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