The solution of the differential equation ` xy^(2) dy - (x^(3) +y^(3))dx = 0 ` is

Options

A. ` y^(3)=3x^(3)+C`
B. `y^(3)=3x^(3) log (Cx)`
C. `y^(3)=3x^(3)+log(Cx)`
D. `y^(3)+3x^(3)=log (Cx)`

Answer

c

Explanation

Given , ` (dy)/(dx) = (x^(3)+y^(3))/(xy^(2))`
Put ` y = vx rArr (dy)/(dx) = v + x (dv)/(dx)`
` :. x (dv)/(dx) + v = (1+v^(3))/(v^(2)) rArr v^(2) dv = (dx)/x`
On integrating both sides , we get
` (v^(3))/3 = log x + log C`
` rArr 1/3 (y/3)^(3) = log x + log C`
` y^(3) = 3x^(3) log Cx`