EGR 509 Advanced Differential Equations __________________ LAST NAME, FIRST Problem set #1

1. The population (P) of mosquitoes in a certain area increases at a rate proportional to the current population and, in the absence of other factors, the population doubles each week. There are 200,000 mosquitoes in the area initially, and predators eat 20,000 mosquitoes/day. Determine the population of mosquitoes in the area at any time.

Ans: P = 201,977.31 1977.31exp[ln(2)t], t in week

2. (Problem 1.11) A spherical water drop loses volume by evaporation at a rate proportional to its surface area. Express its radius at time t in terms of the constant of proportionality and its radius ro at t = 0.

Ans: r = ro t

3. (Problem 1.51) All observations on animal tumors indicate that their sizes obey the Gompertz growth law

= ksln

In this equation k and C are positive constants. By putting y = ln s, determine the size s at a later time given s = so at t = 0.

Ans: s = Cexp

4. (Problem 1.101) Bernoulli’s differential equation has the form

+ f(t)y = g(t)yn

Show that it can be made linear by the substitution z = y1-n. Hence find the general solution of

+ ty = ty2

Ans: y =

5. (Problem 1.121) The function y1(t) is given to be a solution of the Riccati equation

+ a(t)y + b(t)y2 = c(t)

1 Differential Equations and Mathematical Biology by Jones and Sleeman, Chapman & Hall/CRC, 2003

Show that the general solution can be found by putting y = y1 + w and using the substitution z = w1-n. Hence find the general solution of

+ 2 y2 =

given that y1 = 2/t is a solution.

Ans: y = +

6. (Problem 1.151) According to Newton’s law of cooling, the rate of decrease of temperature of a body is proportional to the difference between its temperature and that of its environment. If the temperature of the environment is 20oC and the body cools from 80oC to 60oC in 1 hr, determine the time required for the body to cool to 30oC.

Ans: 4.419 hr

7. (Problem 1.181) The amount of light absorbed by a layer of material is proportional to the incident light and to the thickness of the layer. If a layer 35 cm thick absorbs half the light incident on its surface, what percentage of the incident light will be absorbed by a layer 200 cm thick?

Ans: 98.1% absorbed

8. (Problem 1.191) When a drug is administered, it forms a concentration in the body fluids. This concentration diminishes in time since the drug is metabolized by the liver and eliminated out of the body fluids by the kidney. The simplest equation used to model the reduction of the drug concentration c(t) is given by

=

In this equation is a constant that measures the rapidity at which the concentration falls.

When the drug theophylline is administered for asthma, a concentration below 5 mg liter -1 has little effect and undesirable side-effects appear if the concentration exceeds 20 mg liter -1. For a body that weights W kg, the concentration when D mg dose is administered is 2D/W mg liter-1. If = 6 hr, D = 500 mg, and W = 70 kg, determinea) The elapse time for a second dose to prevent the concentration from becoming ineffective.b) The elapse time after a second dose before a third dose is necessary.c) The shortest safe time interval to at which doses of 500 mg can be given regularly.

Ans: (a) 6.3 hr, (b) 8.0996 hr, (c) 7.5166 hr

9. (Problem 1.1-42) Derive the general solution of = 0 using the following change of

variables

= ax + bt, = cx + dt.

The arbitrary constants a, b, c, d should be chosen so that the PDE can be reduced to an ordinary differential equation.Ans: u(x, t) = f(x + t)

2 Partial Differential Equations and Boundary Value Problems by Nakhle Asmar, Prentice Hall, 2000