using impliciate deferation find y^2+X^3=3XY

Question

This was a homework question and I have a mental block and do not know where to begin

Answer 1

using impliciate deferation find dy/dx for y^2+X^3=3XY

 

y^2 + x^3 = 3xy

differentiating both sides wrt x,

d(y^2)/dx + d(x^3)/dx = 3.d(xy)/dx

Taking the three differentials individually,

d(y^2)/dx = d(y^2)/dy * dy/dx      (using chain rule)

d(y^2)/dx = 2y * y’                           (where y’ = dy/dx)

d(x^3)/dx = 3x^2                              (normal differentiation rule)

3.d(xy)/dx = 3(1.y + x.y’)                  (using product rule)

Summing these three results appropriately,

2y.y’ + 3x^2 = 3(y + x.y’)

2y.y’ + 3x^2 = 3y + 3x.y’

2y.y’ – 3x.y’ = 3y – 3x^2

(2y – 3x).y’ = 3(y – x^2)

y’ = dy/dx = 3(y – x^2)/(2y – 3x)