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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 9: 15E (page 426)

Question:Justify the 鈥淩ule of 69鈥�: If a quantity grows at a constant instantaneous rate of, then its doubling time is about. Example: In 2008 the population of Madagascar was about 20 million, growing at an annual rate of about 3%, with a doubling time of about 69/3 = 23 years.

Short Answer

Expert verified

Answer

The solution is the doubling time is about $\frac{69}{K}$

Step by step solution

01

Definition of the differential equation

Consider the differential equation $\frac{d y}{d x} = k x$ with initial value (k is an arbitrary constant). The solution is $x (t) = x_{0} e^{k t}$ .

The solution of the linear differential equation $\frac{d y}{d x} = k x$ and is $y (0) = C, y = C e^{k x}$ .

02

Step2: Explanation of the solution

Consider the doubling time as follows.

$x (T) = x_{0} e^{\frac{k T}{100}} 2 x_{0} = x_{0} e^{\frac{k T}{100}} 2 = e^{\frac{k T}{100}}$

Now, taking log on both sides as follows.

$2 = e^{\frac{k T}{100}} l n (2) = l n (e^{\frac{k T}{100}}) l n (2) = \frac{k T}{100} 100 l n (2) = k T$

Simplify further as follows.

$100 l n (2) = k T T = \frac{100 l n (2)}{k} T = \frac{100 (0.69)}{k} T 鈮� \frac{69}{k}$

Therefore, if a quantity grows at a constant instantaneous rate of , then its doubling time is about $\frac{69}{k}$ .

Thus, if a quantity grows at a constant instantaneous rate of , then its doubling time is about $\frac{69}{k}$ .

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Most popular questions from this chapter

True or False? If the trace and the determinant of a $3 脳 3$ matrix A are both negative, then the origin is a stable equilibrium solution of the system $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ . Justify your answer.

Solve the initial value problem in $\frac{dx}{dt} + 3 x = 7, x (0) = 0$

Solve the differential equation $f'' (t) + f' (t) - 12 f (t) = 0$ and find solution of the differential equation.

Ngozi opens a bank account with an initial balance of 1,000 Nigerian naira. Let b(t) be the balance in the account at time t; we are told that b(0) = 1,000. The bank is paying interest at a continuous rate of 5% per year. Ngozi makes deposits into the account at a continuous rate of s(t) (measured in naira per year). We are told that s(0) = 1,000 and that s(t) is increasing at a continuous rate of 7% per year. (Ngozi can save more as her income goes up over time.)

(a) set up a linear system of the form $| \begin{matrix} \frac{d b}{d t} = ? b + ? s \\ \frac{d s}{d t} = ? b + ? s \end{matrix} |$

(b) Find $b (t)$ and $s (t)$ .

Consider a quadratic form $q (\overset{鈫�}{x}) = \overset{鈫�}{x} 路 A \overset{鈫�}{x}$ of two variables. Consider the system of differential equations $|\frac{{dx}_{1}}{dt} = \frac{鈭� q}{鈭� x_{1}} \frac{{dx}_{2}}{dt} = \frac{鈭� q}{鈭� x_{2}}|$

Or more sufficiently

$\frac{d \overset{鈫�}{x}}{d t} = g r a d q$

Show that the system $\frac{d \overset{鈫�}{x}}{d t} = g r a d q$ is linear by finding a matrix B(in terms of the symmetric matrix A) such that $g r a d q = B \overset{鈫�}{x}$
When q is negative definite,draw a sketch showing possible level curves of q. On the same sketch draw a few trajectories of the system localid="1662089983392">dx鈫�dt=gradq.What does your sketch suggest about the stability of the systemlocalid="1662089999708">dx鈫�dt=gradq
Do the same as in part b for an indefinite quadratic form
Explain the relationship between the definiteness of the form q and the stability of the systemlocalid="1662090018926">dx鈫�dt=gradq

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