Solve the differential equation: dy/dx= e^-2x/1+e^-2x. Write in the form y= f(x)

Question

Solve the differential equation: dy/dx= e^-2x/1+e^-2x. Write in the form y= f(x)

Answer 1

dy = exp(-2x)dx / [ 1 + exp(-2x)

integrates both sides:

U = 1 + exp(-2x)

dU = (-1/2) exp(-2x) dx

dy = (-1/2) dU/U

y = (-1/2) ln |U|

= (1/2) ln | 1 + exp(-2x)| + c

Answer 2

Integrate both sides

y = ∫e^-2x/(1+e^-2x)dx

let u = 1+e^-2x, du = -2e^-2xdx

-(1/2)∫du/u = -ln(u)/2 + C = -ln(1+e^-2x)/2 + C

y = -ln(1+e^-2x)/2 + C