Initial Value Problems As you saw in Engineering Calculus I, there are many situ
ID: 3142823 • Letter: I
Question
Initial Value Problems As you saw in Engineering Calculus I, there are many situations in science and engineering for which information about a rate of change is known. This information may be expressed asa differential equation (that is, an equation that contains one or more derivatives). An initial value problem consists of a differential equation and a known value or values of the dependent variable, expressed as an initial condition. Population Models Recall that a variable y is said to be directly proportional to another variable x if the ratio of their values is always the same, that is k, where k is called the constant of proportionality. This relationship is more commonly written as y kr In some cases, the rate of change of a population P is directly proportional to the value of the population at any time. This can be expressed as dP/ dt-k or dP dt The constant of proportionality k equals the fractional growth rate of the population. This last equation is called a differential equation, because it contains a derivative. But unlike simpler differential equations you may have previously solved, this equation cannot be directly integrated to find P), because the population P appears on the right-hand side in the expression for its own derivative. dP dt However, the equation -kP is a separable differential equation, because it can be algebraically rewritten so the variables are separated in such a way that both sides of the equation can be integrated to produce the function P(). A Model for World Population Growth In 1990, the population of the Earth was 5.3 billion. At that time, the birth rate was 27.4 per housand and the death rate was 14.6 per thousand. per oc . What was the fractional net growth rate k for world population in 1990, written as a decimal? . Write a differential equation that models world population growth for the years after 1990, ssuming a constant growth rateExplanation / Answer
a)
dP/dT = kP
k - function net growth of world population
k = birth rate - death rate = 27.4 - 14.6 = 12.8 per thousand = 12. 8 x 106 per billion = 0.0128
b)
dP/dt = kP
k - function net growth of world population = 12. 8 x 106 per billion = 0.0128
P - world population at time t
t - time
c)
at t = 0, P (t=0) = 5.3
d)
dP/P = k dt
e)
integrating both side
ln (P) = kt + C
since C is anway constant, having 2 constants both side just increases the complexity and requirement of an additional intial condition. So instead of having C1 on left and C2 on right, we can have C = C2 - C1 on right.
f)
at t = 0 , P = 5.3 so
C = ln (5.3)
so equation become
ln (P) = .0128 t + ln (5.3)
(g)
ln (P/5.3) = .0128 t
P = 5.3 e(0.0128t)
t is 27 for year 2017
P (2017) = 7.488038 Billion
It is fairly close to accurate number of 7.5 billion
(h)
ln (10/5.3) = 0.0128 t
t = 50
so by 2040 we will have 10 billion