Solve the following problem step by step

Question

Find the solution of (1+ e^x)yy' = e^x that passes through (0, 1).

Answer 1

(1+e^x)ydy/dx = e^x

Use separation of variables

∫ydy = ∫e^x/(1+e^x) dx

y²/2 = ln(1+e^x) + C

At (0,1)

1/2 = ln(2) + C

C = 1/2 - ln(2)

y²/2 = ln(1+e^x) -ln(2) + 1/2 = ln((1+e^x)/2) + 1/2

y = ±√( 2ln((1+e^x)/2) + 1)