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AP EAMCET202418 May 2024Morning ShiftMathematicsDifferential EquationsActual

The differential equation representing the family of circles having their centres on Y-axis is (y₁= d y d x . and .y₂= d^2 y d x^2 )

Options

  1. Ay₂=y (y₁^2+1 )
  2. By₂=x y (y₁^2+1 )
  3. Cx y₂=y₁ (y₁^2+1 )
  4. Dx y₂=y (y₁^2+1 )

Correct answer

C. x y₂=y₁ (y₁^2+1 )

Step-by-step solution

Let a point on y -axis be (0, b) On equation of circle have centre (0, b) and radius a is x^2+(y-b)^2=a^2 Now, 2 x+2(y-b) ( d y d x )=0 (i) again differentiating 1+(y- b ) d^2 y d x^2 + ( d y d x )^2=0 y-b= -y₁^2-1 y₂ by (i), x+ (-y₁^2-1 ) y₁ y₂ =0 x y₂=y₁ (1+y₁^2 )

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