Solving dP/dt=k1(P)-k2(P) for different constant relationships

[andrewdavid]

I have this population differential equation dP/dt=k1(P)-k2(P) where k1 and k2 are proportionality constants. I need to integrate and analyze where k1>k2, k1=k2, and k1<k2. Trouble is, I don't think I'm integrating this right. I get P=e^(t+C)(k1-k2). I know this should be easy but I don't think it's right. Little help?

 

[Data]

the solution is

$$P(t) = Ae^{(k_1-k_2)t}$$

for some constant $A$, which might be equivalent to yours, or it might not (I can't tell whether you mean that $k_1-k_2$ is in the exponent or not. If it is, then yours is fine).

 

[dextercioby]

Yes,the jonction between the 2 formulae is made simply

$$P(t)=e^{\left[\left(k_{1}-k_{2}\right)(t+C)\right]}=e^{C\left(k_{1}-k_{2}\right)}e^{\left(k_{1}-k_{2}\right) t} =A e^{\left(k_{1}-k_{2}\right) t}$$

,where i defined

$$A=:e^{C\left(k_{1}-k_{2}\right)}$$

Daniel.