Question

y dx + (2xy - e^(-2y) dy) = 0
solve using integrating factor

Answer 1

Divide by dy then y and rearrange:
x'+2x=e^(-2y)/y
Integrating factor: e^integral(2dy) = e^(2y)
(x*e^(2y))' = e^(2y)*e^(-2y)/y = 1/y
Integrate:
x*e^(2y) = ln(y) + c

Solution: x(y) = (ln(y) + c) * e^(-2y)