Find an integrating factor and solve the given equation (3x + 6/y) + (x^2/y + 3y/x)dy/dx =0
(3x + 6/y)dx + (x^2/y + 3y/x)dy =0
𝑀𝑦 = -6/y^2 N𝑥=2x/y -3y/x^2 not exact
(N𝑥-𝑀𝑦) / (x*M-N*y) = (2x/y - 3y/x^2 + 6/y^2)/(2x^2 + 6x/y - 3y^2/x) = 1/xy
Let t represent xy
μ = e^∫1/t𝑑t = e^ln|t| = t = xy
multiply xy on both side
(3yx^2 + 6x)dx + (x^3 +3y^2)dy = 0
𝜙(𝑥,𝑦)=∫3yx^2 + 6x𝑑𝑥=yx^3 + 3x^2 + ℎ(𝑦)
𝜙(𝑦)=x^3 + ℎ′(𝑦) = N
ℎ′(𝑦)=3y^2
ℎ(𝑦) = y^3
𝜙(𝑥,𝑦)=yx^3 + 3x^2 + y^3 = c
