Solving the Tumor Growth Problem: Find B_0, a & More

[missteresadee]

Problem: A tumor may be regarded as a population of multiplying cells. It is found empirically that the "birth rate" of cells in a tumor decreases exponentially with time, so that birth rate is given by B(t)=(B_0)*e^(-a*t) where B_0 and a are constants. Consequently, the governing equation for the tumor population P(t) at any time t is
dP/dt=(B_o)e^(-at)P

Solve this differential equation exactly. Suppose the initially at t=0, there are 10^6 tumor cells, and that P(t) is then increase at a rate of 3*10^5 cells per month. After 6 months, the tumor has double in population. Find a, and find the behavior of the tumor population as t goes to infinity

I know from the given information, I can find B_0 by using this equation
3*10^5=(B_0)*e^(-a*0)*10^6
given rate of change is 3*10^5; t=0, and p=10^6
B_0=3/10

I am having trouble finding my constant C, a...

 

[pasmith]

Problem: A tumor may be regarded as a population of multiplying cells. It is found empirically that the "birth rate" of cells in a tumor decreases exponentially with time, so that birth rate is given by B(t)=(B_0)*e^(-a*t) where B_0 and a are constants. Consequently, the governing equation for the tumor population P(t) at any time t is
dP/dt=(B_o)e^(-at)P

Solve this differential equation exactly. Suppose the initially at t=0, there are 10^6 tumor cells, and that P(t) is then increase at a rate of 3*10^5 cells per month. After 6 months, the tumor has double in population. Find a, and find the behavior of the tumor population as t goes to infinity

I know from the given information, I can find B_0 by using this equation
3*10^5=(B_0)*e^(-a*0)*10^6
given rate of change is 3*10^5; t=0, and p=10^6
B_0=3/10

I am having trouble finding my constant C, a...

What do you get if you solve the differential equation to work out P(6 months) in terms of $a$, and set that equal to $2P_0$?

 

[missteresadee]

Yup! I figured it out, I had two system of equations with two unknowns which I can solve =) I was trying to use maple program to solve this equation, and finally figured out how to display the answers in approximation instead of weird exact numbers. THANKS!

 

[Listiba]

I'm assuming I'm in your class (lol).

Can you explain how you were able to solve this? I tried solving for B0, got 3/10. Tried solving for P, got e^(-Be^(-at))/a

Plugging B and 2P into the equation for P gives a lambertW function which I'm assuming is far beyond the scope of DE.