7 10 2
Find the sum of eigen values of A = 0 2 0
2 05
Answers
Step-by-step explanation:
du
dt = 5u, u(0) = −3, (b)
du
dt = 2u, u(1) = 3, (c)
du
dt = −3u, u(−1) = 1.
Solution: (a) u(t) = −3e
5t
, (b) u(t) = 3e
2(t−1)
, (c) u(t) = e
−3(t+1)
.
8.1.2. Suppose a radioactive material has a half-life of 100 years. What is the decay rate γ?
Starting with an initial sample of 100 grams, how much will be left after 10 years? 100
years? 1, 000 years?
Solution: γ = log 2/100 ≈ 0.0069. After 10 years: 93.3033 gram; after 100 years 50 gram; after
1000 years 0.0977 gram.
8.1.3. Carbon-14 has a half-life of 5730 years. Human skeletal fragments discovered in a cave
are analyzed and found to have only 6.24% of the carbon-14 that living tissue would have.
How old are the remains?
Solution: Solve e
−(log 2)t/5730 = .0624 for t = −5730 log .0624/ log 2 = 22, 933 years.
8.1.4. Prove that if t
?
is the half-life of a radioactive material, then u(nt
?
) = 2
−n u(0). Ex-
plain the meaning of this equation in your own words.
Solution: By (8.6), u(t) = u(0) e
−(log 2)t/t?
= u(0) “
1
2
”t/t?
= 2
−n u(0) when t = nt
?
. After
every time period of duration t
?
, the amount of material is cut in half.
8.1.5. A bacteria colony grows according to the equation du/dt = 1.3 u. How long until the
colony doubles? quadruples? If the initial population is 2, how long until the population
reaches 2 million?
Solution: u(t) = u(0) e
1.3t
. To double, we need e
1.3t = 2, so t = log 2/1.3 = 0.5332. To
quadruple takes twice as long, t = 1.0664. To reach 2 million needs t = log 106
/1.3 = 10.6273.
8.1.6. Deer in Northern Minnesota reproduce according to the linear differential equation du
dt =
.27u where t is measured in years. If the initial population is u(0) = 5, 000 and the environ-
ment can sustain at most 1, 000, 000 deer, how long until the deer run out of resources?
Solution: The solution is u(t) = u(0) e
.27t
. For the given initial conditions, u(t) = 1, 000, 000
when t = log(1000000/5000)/.27 = 19.6234 years.
♦ 8.1.7. Consider the inhomogeneous differential equation du
dt = au + b, where a, b are constants.
(a) Show that u? = −b/a is a constant equilibrium solution. (b) Solve the differential
equation. Hint: Look at the differential equation satisfied by v = u − u?
. (c) Discuss the
stability of the equilibrium solution u?
.
Solution:
evv 9/9/04 415 °c 2004 Peter J. Olver