# problem solving in differential equations

• Feb 16th 2009, 10:20 AM
mathprincess24
problem solving in differential equations
This one has been bugging me. I cant seem to figure it out. can anyone help..??

A tumor may be regarded as a population of multiplying cells. It is found empirically that the birth rate of the cells in a tumor decreases exponentially with time so that β( t) = β0e-αt ( where α and β0 are positive constants).
And hence
dP/dt = β0e-αt P
P(0) = P0

1.Solve this initial value problem for P(t) = P0exp ( ß0/α(1- e-αt)

Observe that P(t) approaches the finite limiting population P0exp(ß0/α) as t approaches infinity.
2.Suppose that at time t=0 there are P0 = 106 cells and that P(t) is then increasing at the rate of 3 x105 cells per month. After 6 months the tumor has doubled (in size and in number of cells). Solve numerically for α , and then find the limiting population of the tumor.
• Feb 16th 2009, 10:55 AM
HallsofIvy
Quote:

Originally Posted by mathprincess24
This one has been bugging me. I cant seem to figure it out. can anyone help..??

A tumor may be regarded as a population of multiplying cells. It is found empirically that the birth rate of the cells in a tumor decreases exponentially with time so that β( t) = β0e-αt ( where α and β0 are positive constants).
And hence
dP/dt = β0e-αt P
P(0) = P0

1.Solve this initial value problem for P(t) = P0exp ( ß0/α(1- e-αt)

Observe that P(t) approaches the finite limiting population P0exp(ß0/α) as t approaches infinity.
2.Suppose that at time t=0 there are P0 = 106 cells and that P(t) is then increasing at the rate of 3 x105 cells per month. After 6 months the tumor has doubled (in size and in number of cells). Solve numerically for α , and then find the limiting population of the tumor.

What exactly is your difficulty? T start with, do you know how to solve a separable differential equation such at this?
• Feb 16th 2009, 10:59 AM
mathprincess24
the letters just make it confusing
• Feb 16th 2009, 02:02 PM
mathprincess24
am i solving the equation written under question 1 or am i solving the one given in the introductory information. i just dont even know where to start with this. can you help me a bit more?
• Feb 17th 2009, 04:00 PM
mathprincess24
this is still troubling me. can anyone help me with this?