Homework for §2.1
Solve the initial value problems below.
dx
= 1 − x2 , x(0) = 3.
dt
dx
2.
= 9 − 4x2 , x(0) = 0.
dt
.............................................................................
1.
3. The rate of change of a population P is proportional to the square root of P . At
time t = 0 (months) the population numbers 100 rabbits and is increasing at the
rate of 20 rabbits per month. How many rabbits will there be one year later?
4. 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 β(t) = β0 e−αt (where α and β0 are positive constants), and hence
dP
= β0 e−αt P, P (0) = P0 .
dt
Solve this initial value problem for
P (t) = P0 exp
β0
−αt
(1 − e ) .
α
Observe that P (t) approaches the finite limiting population P0 exp(β0 /α) as t →
+∞.
5. Derive the solution
P (t) =
M P0
P0 + (M − P0 )e−kM t
of the logistic initial value problem P 0 = kP (M − P ), P (0) = P0 .