(2y + xy)dx + 2xdy = 0
rearranging,
dy/dx = -(2y + xy)/(2x) = -(2 + x)/(2x).y = -g(x).y
dy/dx + g(x).y = 0
Integrating Factor
IF = exp(int g(x) dx)
int g(x) dx = int (2 + x)/(2x) dx = int {1/x + 1/2} dx
int g(x) dx = ln(x) + x/2
Hence,
IF = exp(ln(x) + x/2) = x*e^(x/2)
The DE now becomes,
d(IF*y) = 0
d(x*e^(x/2).y) = 0
Integrating,
x*e^(x/2).y = const
Answer: xy.e^(x/2) = c