dy/dx+((2x+1)/x)y=e^-2x

Answer 1

This is a good example a first-order linear d.e.

y' + P(x)y = Q(x)
where
P(x) = (2x+1)/x
Q(x) = e^-2x

Int(2x+1)/x dx = Int(2 + 1/x dx) = 2x + lnx
e^(2x+lnx) = xe^2x

so an integrating factor is xe^2x, giving us

xe^2x*y' + xe^2x * (2x+1)/x y = xe^2x * e^-2x

d/dx(xe^2x y) = x
so,
xe^2x y = x^2/2 + C

y = e^-2x (x/2 + C/x)

Answer 2

Great! Thank you I understand now :)