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).
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.