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AP EAMCET2008MathematicsDifferential Equations

The solution of the differential equation x y^2 d y- (x^3+y^3 ) d x=0 is

Options

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

Correct answer

B. y^3=3 x^3 (c x)

Step-by-step solution

Given differential equation can be rewritten as d y d x = x^3+y^3 x y^2 It is a homogeneous differential equation. Put y=v x d y d x =v+x d v d x array rlrl & & x d v d x +v & = x^3+v^3 x^3 x^3 v^2 & x d v d x +v & = 1+v^3 v^2 & x d v d x & = 1 v^2 & v^2 d v & = d x x array On integrating both sides, we get aligned v^3 3 & = x+ c 1 3 ( y x )^3 & = x+ c y^3 & =3 x^3 c x aligned

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