Solving a first order differential equation

[zak8000]

hi

the differential equation i am attempting to solve is:

$$\frac {dP} {dx} = \frac {gP} {1+P/Psat}$$

here is what I have done:

$$\frac {dP} {dx} = \frac {gP*Psat} {Psat+P}$$

divide both sides by $$\frac {Psat+P} {gP*Psat}$$

to get:
$$\frac {Psat+P} {P*Psat} \frac {dP} {dx} =g$$

$$\int \frac {Psat+P} {P*Psat} dp = \int gdx$$

$$\int \frac {dp} {P}+ \int \frac {dp} {Psat} =gx+c$$
$$ln(P)+ \frac {P} {Psat} =gx+c$$

now how do i rearrange P on one side with everything else on the other side

 

[Sourabh N]

By realizing, P = ln(e^P)

 

[JJacquelin]

Hi zak8000 !

In practice, the equation ln(P)+P/Psat = gx+e is solved thanks to numerical methods.
The analytic solution requieres a special function W(X), namely the Lambert W function.
P/Psat = W(X) with X=exp(gx+e)/Psat