How to you find the general solution of #dy/dx=e^x/(1+e^x)#?

Use the separation of variables method:

Integrate both sides:

To find the general solution of dy/dx = e^x / (1 + e^x), first, rewrite the equation as:

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

Next, integrate both sides:

∫(dy / (1 + e^x)) = ∫e^x dx

This integral can be solved using the substitution method. Let u = 1 + e^x, then du = e^x dx. Substituting these values:

∫(1/u) du = ∫dx

Integrating both sides:

ln|u| + C₁ = x + C₂

Where C₁ and C₂ are constants of integration.

Substitute back for u:

ln|1 + e^x| + C₁ = x + C₂

Exponentiate both sides:

|1 + e^x| = e^(x + C₂ - C₁)

Since e^(C₂ - C₁) is a constant, let K = e^(C₂ - C₁):

|1 + e^x| = Ke^x

This gives us two cases to consider:

1. If 1 + e^x > 0:

1 + e^x = Ke^x

2. If 1 + e^x < 0:

-(1 + e^x) = Ke^x

Now solve each case for y:

1. If 1 + e^x > 0:

y = ln(1 + Ke^x)

2. If 1 + e^x < 0:

y = ln(-1 - Ke^x)

These are the general solutions for the differential equation dy/dx = e^x / (1 + e^x), where K is any nonzero constant.