# Is there an analytical solution to dCs/dt = gCs(1-Cs/Csmax) - K*(X/Y + c/exp(Yt))? (nonlinear, nonexact)

#### amcqueen101

##### New member
dCs/dt = gCs(1-Cs/Csmax) - K*(X/Y + c/exp(Yt));

Hello,

The above equation is the one in which I am struggling with, I was curious is this could be solved analytically at all? (I have a numerical solution to the problem). The initial condition for Cs is 0.7 (as it stands) but may change. All other variables (except t) are just constants.

As an engineer I only recall learning how to solve first order ODEs using integrating factor or by separation. The former doesn't hold as the problem is nonlinear and the latter doesn't hold as the equation isn't separable.

If anyone could help, would be greatly appreciated.

Ally

#### HallsofIvy

##### Elite Member
Is "Csmax" Cs times "max" or is it a maximum value of Cs? If the latter is it a given constant or is it to be calculated from the problem?

#### Subhotosh Khan

##### Super Moderator
Staff member
dCs/dt = gCs(1-Cs/Csmax) - K*(X/Y + c/exp(Yt));

Hello,

The above equation is the one in which I am struggling with, I was curious is this could be solved analytically at all? (I have a numerical solution to the problem). The initial condition for Cs is 0.7 (as it stands) but may change. All other variables (except t) are just constants.

As an engineer I only recall learning how to solve first order ODEs using integrating factor or by separation. The former doesn't hold as the problem is nonlinear and the latter doesn't hold as the equation isn't separable.

If anyone could help, would be greatly appreciated.

Ally
Is c different from Cs?

#### amcqueen101

##### New member
Is "Csmax" Cs times "max" or is it a maximum value of Cs? If the latter is it a given constant or is it to be calculated from the problem?

Hi,

Csmax is a constant, it’s the carrying capacity of the cells species, Cs

#### amcqueen101

##### New member
yeah, c is different from Cs.

It is a constant of integration calculated when solving the analytical solution to a different equation.