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MHT CET202526 Apr 2025Morning ShiftMathematicsDifferential EquationsActual

The differential equation which represents the family of curves y = C ₁ e ^ c ₂ x , where C ₁, C ₂ are arbitrary constants is

Options

  1. Ay^ =y^ y
  2. By y^ =y^
  3. Cy y^ = (y^ )^2
  4. Dy ^ = y ^2

Correct answer

C. y y^ = (y^ )^2

Step-by-step solution

The family of curves y = C₁ e^ c₂ x contains two arbitrary constants C₁ and C₂ , requiring a second-order differential equation to eliminate them. Differentiating y = C₁ e^ c₂ x gives y' = C₁ c₂ e^ c₂ x , and differentiating again yields y'' = C₁ c₂^2 e^ c₂ x . From the first derivative, y' = c₂ y since C₁ e^ c₂ x = y , giving c₂ = y' y . Similarly, y'' = c₂ y' by substituting C₁ c₂ e^ c₂ x = y' into the second derivative. Substituting c₂ = y' y into y'' = c₂ y' produces y'' = (y')^2 y , which rearranges to the dif

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