1550078481-Ordinary_Differential_Equations__Roberts_

Applications of the Initial Value Problem y' = f(x, y); y(c) = d 161

concentrations, and if k is the velocity constant, then y(t) satisfies the

initial value problem

Suppose k = 1.5 liter/(mole·sec), CA(O) = 9 moles/liter, and CB(O) =

5 moles/liter.

a. Find and graph y(t) on the interval [O, 5].

b. Find limt_, 00 y(t).

c. What is the limiting concentrations of the reactants A and B?

MISCELLANEOUS EXERCISES

1. The error function is usually defined as

2 r 2

erf(x) = fa Jo e-t dt.

This definition of the error function as an integral is equivalent to the

initial value problem

I^2 e -x2

y = fa ; y(O) = 0.

Solve this initial value problem on the interval [O, 5] and graph the so-

lution.

2. Suppose the pollution index expressed in parts per million over a 24 hour
   period satisfies the differential equation

dP .34 - .04t

dt v'50 + 25t - t^2

At 12 midnight (t = 0) the pollution index is found to be .43 parts per

million. Compute and graph the pollution index for the next 24 hours.

At what time is the pollution index the highest? lowest?

3. Let T be the absolute temperature of a body and let A be the absolute
   temperature of the surrounding medium. According to Stefan's law
   of radiation the rate of change of the temperature of the body is pro-
   portional to the difference between the fourth power of the temperature
   of the body and the fourth power of the temperature of the surround-
   ing medium. Thus, according to Stefan's law dT/dt = -k(T^4 - A^4 ).
   Compute and graph on the interval [O, 20] the temperature T of a body
   whose initial temperature is 3000° K, whose constant of proportionality