y(2 x y+e^x)dx=e^xdy

Question

type of bifferential equation

Answer 1

y(2xy+e^x)dx=e^xdy

2xy^2 + e^x.y = e^x.y'

e^x.y = -2xy^2 + e^x.y'

e^x = -2xy + e^x.y'/y

looking at the rhs, -2xy + e^x.y'/y, this is of the form g'(x).y + g(x).y'

where g'(x) = -2x and g(x) = e^x/y

integrating the g'(x), we get g(x) = c - x^2

but g(x) = e^x/y, hence e^x/y = c - x^2

so, y(x) = e^x/(c - x^2)