Chapter 2

Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the recursion of (3) by hand, first using \(h=0.1\) and then using \(h=0.05\). $$ y^{\prime}=2 x-3 y+1, \quad y(1)=5 ; y(1.2) $$

The population of a community is known to increase at a rate proportional to the number of people present at time \(t\). If an initial population \(P_{0}\) has doubled in 5 years, how long will it take to triple? To quadruple?

The number \(N(t)\) of supermarkets throughout the country that are using a computerized checkout system is described by the initial-value problem$$\frac{d N}{d t}=N(1-0.0005 N), \quad N(0)=1.$$ (a) Use the phase portrait concept of Section \(2.1\) to predict how many supermarkets are expected to adopt the new procedure over a long period of time. By hand, sketch a solution curve of the given initial-value problem. (b) Solve the initial-value problem and then use a graphing utility to verify the solution curve in part (a). How many companies are expected to adopt the new technology when \(t=10 ?\)

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ \frac{d y}{d x}=5 y $$

Solve the given differential equation by using an appropriate substitution. $$ (x-y) d x+x d y=0 $$

Determine whether the given differential equation is exact. If it is exact, solve it. $$ (2 x-1) d x+(3 y+7) d y=0 $$

\(\frac{d y}{d x}=\sin 5 x\)

In Problems \(1-10\), solve the given differential equation by using an appropriate substitution. $$ (x-y) d x+x d y=0 $$

Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the recursion of (3) by hand, first using \(h=0.1\) and then using \(h=0.05\). $$ y^{\prime}=x+y^{2}, \quad y(0)=0 ; y(0.2) $$

The number \(N(t)\) of people in a community who are exposed to a particularadvertisement is governed by the logisticequation. Initially \(N(0)=500\), and it is observed that \(N(1)=1000\). Solve for \(N(t)\) if it is predicted that the limiting number of people in the community who will see the advertisement is 50,000 .

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ \frac{d y}{d x}+2 y=0 $$

Solve the given differential equation by using an appropriate substitution. $$ (x+y) d x+x d y=0 $$

Determine whether the given differential equation is exact. If it is exact, solve it. $$ (2 x+y) d x-(x+6 y) d y=0 $$

In Problems \(1-10\), solve the given differential equation by using an appropriate substitution. $$ (x+y) d x+x d y=0 $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ \frac{d y}{d x}=(x+1)^{2} $$

The population of a town grows at a rate proportional to the population present at time \(t\). The initial population of 500 increases by \(15 \%\) in 10 years. What will the population be in 30 years? How fast is the population growing at \(t=30 ?\)

A model for the population \(P(t)\) in a suburb of a large city is given by the initial-value problem $$ \frac{d P}{d t}=P\left(10^{-1}-10^{-7} P\right), \quad P(0)=5000, $$ where \(t\) is measured in months. What is the limiting value of the population? At what time will the population be equal to one-half of this limiting value?

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ \frac{d y}{d x}+y=e^{3 x} $$

Solve the given differential equation by using an appropriate substitution. $$ x d x+(y-2 x) d y=0 $$

Determine whether the given differential equation is exact. If it is exact, solve it. $$ (5 x+4 y) d x+\left(4 x-8 y^{3}\right) d y=0 $$

In Problems \(1-10\), solve the given differential equation by using an appropriate substitution. $$ x d x+(y-2 x) d y=0 $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ d x+e^{3 x} d y=0 $$

Reproduce the given computer-generated direction field. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points. Use different colored pencils for each solution curve. $$ \frac{d y}{d x}=1-x y $$ (a) \(y(0)=0\) (b) \(y(-1)=0\) (c) \(y(2)=2\) (d) \(y(0)=-4\)

The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time \(t\). After 3 hours it is observed that 400 bacteria are present. After 10 hours 2000 bacteria are present. What was the initial number of bacteria?

(a) Census data for the United States between 1790 and 1950 is given in the following table. Construct alogistic population model using the data from 1790,1850 , and 1910 . $$ \begin{array}{lr} \hline \text { Year } & \text { Population (in millions) } \\ \hline 1790 & 3.929 \\ 1800 & 5.308 \\ 1810 & 7.240 \\ 1820 & 9.638 \\ 1830 & 12.866 \\ 1840 & 17.069 \\ 1850 & 23.192 \\ 1860 & 31.433 \\ 1870 & 38.558 \\ 1880 & 50.156 \\ 1890 & 62.948 \\ 1900 & 75.996 \\ 1910 & 91.972 \\ 1920 & 105.711 \\ 1930 & 122.775 \\ 1940 & 131.669 \\ 1950 & 150.697 \\ \hline \end{array} $$ (b) Construct a table comparing actual census population with the population predicted by the model in part (a). Compute the error and the percentage error for each entry pair.

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ 3 \frac{d y}{d x}+12 y=4 $$

Solve the given differential equation by using an appropriate substitution. $$ y d x=2(x+y) d y $$

Determine whether the given differential equation is exact. If it is exact, solve it. $$ (\sin y-y \sin x) d x+(\cos x+x \cos y-y) d y=0 $$

\(d y-(y-1)^{2} d x=0\)

Use Euler's method to obtain a four-decimal approximation of the indicated value. First use \(h=0.1\) and then use \(h=0.05\). Find an explicit solution for each initial-value problem and then construct tables similar to Tables \(2.6 .3\) and 2.6.4. $$ y^{\prime}=2 x y, \quad y(1)=1 ; y(1.5) $$

In Problems \(1-10\), solve the given differential equation by using an appropriate substitution. $$ y d x=2(x+y) d y $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ d y-(y-1)^{2} d x=0 $$

Use a numerical solver and Euler's method to obtain a four-decimal approximation of the indicated value. First use \(h=0.1\) and then use \(h=0.05\). \(-y\) $$ y^{\prime}=e^{-y}, \quad y(0)=0 ; y(0.5) $$

The radioactive isotope of lead, \(\mathrm{Pb}-209\), decays at a rate proportional to the amount present at time \(t\) and has a half-life of \(3.3\) hours. If 1 gram of this isotope is present initially, how long will it take for \(90 \%\) of the lead to decay?

Solve the given differential equation by using an appropriate substitution. $$ \left(y^{2}+y x\right) d x-x^{2} d y=0 $$

Determine whether the given differential equation is exact. If it is exact, solve it. $$ \left(2 x y^{2}-3\right) d x+\left(2 x^{2} y+4\right) d y=0 $$

\(x \frac{d y}{d x}=4 y\)

Use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. \(y^{\prime}=x\) (a) \(y(0)=0\) (b) \(y(0)=-3\)

The number 0 is a critical point of the autonomous differential equation \(d x / d t=x^{n}\), where \(n\) is a positive integer. For what values of \(n\) is 0 asymptotically stable? Semi-stable? Unstable? Repeat for the equation \(d x / d t=-x^{n}\).

Potassium-40Decay The mineral potassium, whose chemical symbol is K, is the eighth most abundant element in the Earth's crust, making up about \(2 \%\) of it by weight, and one of its naturally occurring isotopes, \(\mathrm{K}-40\), is radioactive. The radioactive decay of \(K-40\) is more complex than that of carbon-14 because each of its atoms decays through one of two different nuclear decay reactions into one of two different substances: the mineral calcium- \(40(\mathrm{Ca}-40)\) or the gas argon- \(40(\mathrm{Ar}-40)\). Dating methods have been developed using both of these decay products. In each case, the age of a sample is calculated using the ratio of two numbers: the amount of the parent isotope \(\mathrm{K}-40\) in the sample and the amount of the daughter isotope (Ca-40 or Ar-40) in the sample that is radiogenic, in other words, the substance which originates from the decay of the parent isotope after the formation of the rock. An igneous rock is solidified magma The amount of \(K-40\) in a sample is easy to calculate. \(K-40\) comprises \(1.17 \%\) of naturally occurring potassium, and this small percentage is distributed quite uniformly, so that the mass of \(\mathrm{K}-40\) in the sample is just \(1.17 \%\) of the total mass of potassium in the sample, which can be measured. But for several reasons it is complicated, and sometimes problematic, to determine how much of the \(\mathrm{Ca}-40\) in a sample is radiogenic. In contrast, when an igneous rock is formed by volcanic activity, all of the argon (and other) gas previously trapped in the rock is driven away by the intense heat. At the moment when the rock cools and solidifies, the gas trapped inside the rock has the same composition as the atmosphere. There are three stable isotopes of argon, and in the atmosphere they occur in the following relative abundances: \(0.063 \% \mathrm{Ar}-38\), \(0.337 \%\) Ar-36, and \(99.60 \%\) Ar-40. Of these, just one, Ar-36, is not created radiogenically by the decay of any element, so any Ar- 40 in excess of \(99.60 /(0.337)=295.5\) times the amount of Ar-36 must be radiogenic. So the amount of radiogenic Ar-40 in the sample can be determined from the amounts of Ar-38 and Ar-36 in the sample, which can be measured. Assuming that we have a sample of rock for which the amount of K-40 and the amount of radiogenic Ar- 40 have been determined, how can we calculate the age of the rock? Let \(P(t)\) be the amount of \(\mathrm{K}-40, A(t)\) the amount of radiogenic Ar- 40 , and \(C(t)\) the amount of radiogenic Ca-40 in the sample as functions of time \(t\) in years since the formation of the rock. Then a mathematical model for the decay of \(\mathrm{K}-40\) is the system of linear first-order differential equations $$ \begin{aligned} \frac{d A}{d t} &=\lambda_{A} P \\ \frac{d C}{d t} &=\lambda_{C} P \\ \frac{d P}{d t} &=-\left(\lambda_{A}+\lambda_{C}\right) P \end{aligned} $$ where \(\lambda_{A}=0.581 \times 10^{-10}\) and \(\lambda_{C}=4.962 \times 10^{-10}\). (a) From the system of differential equations find \(P(t)\) if \(P(0)=P_{0}\). (b) Determine the half-life of K- 40 . (c) Use \(P(t)\) from part (a) to find \(A(t)\) and \(C(t)\) if \(A(0)=0\) and \(C(0)=0\). (d) Use your solution for \(A(t)\) in part (c) to determine the percentage of an initial amount \(P_{0}\) of K-40 that decays into Ar- 40 over a very long period of time (that is, \(t \rightarrow \infty\) ). What percentage of \(P_{0}\) decays into \(\mathrm{Ca}-40\) ?

(a) If a constant number \(h\) of fish are harvested from a fishery per unit time, then a model for the population \(P(t)\) of the fishery at time \(t\) is given by $$ \frac{d P}{d t}=P(a-b P)-h, \quad P(0)=P_{0}, $$ where \(a, b, h\), and \(P_{0}\) are positive constants. Suppose \(a=5, b=1\), and \(h=4\). Since the \(\mathrm{DE}\) is autonomous, use the phase portrait concept of Section \(2.1\) to sketch representative solution curves corresponding to the cases \(P_{0}>4,1

The radioactive isotope of lead, Pb-209, decays at a rate proportional to the amount present at time \(t\) and has a half-life of 3.3 hours. If 1 gram of this isotope is present initially, how long will it take for \(90 \%\) of the lead to decay?

In Problems \(1-10\), solve the given differential equation by using an appropriate substitution. $$ \left(y^{2}+y x\right) d x-x^{2} d y=0 $$

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ y^{\prime}+3 x^{2} y=x^{2} $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ x \frac{d y}{d x}=4 y $$

Use a numerical solver and Euler's method to obtain a four-decimal approximation of the indicated value. First use \(h=0.1\) and then use \(h=0.05\). \(-y\) $$ y^{\prime}=x^{2}+y^{2}, \quad y(0)=1 ; y(0.5) $$

Initially, 100 milligrams of a radioactive substance was present. After 6 hours the mass had decreased by \(3 \%\). If the rate of decay is proportional to the amount of the substance present at time \(t\), find the amount remaining after 24 hours.

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ y^{\prime}+2 x y=x^{3} $$

Solve the given differential equation by using an appropriate substitution. $$ \left(y^{2}+y x\right) d x+x^{2} d y=0 $$

Determine whether the given differential equation is exact. If it is exact, solve it. $$ \left(2 y-\frac{1}{x}+\cos 3 x\right) \frac{d y}{d x}+\frac{y}{x^{2}}-4 x^{3}+3 y \sin 3 x=0 $$