Chapter 2

We will describe several methods that are applicable to a certain subclass of first order equations.

Linear equations, separable equations and exact equations are the most important of these.
Some of the important applications of first order differential equations, introduction to the idea of approximating a solution by numerical computation, and some theoretical questions related to existence and uniqueness of solutions are some of the sections of this chapter.
The final section deals with first order difference equations, which have some important points of similarity with differential equations and are easier to investigate.

An equation of this form describes the motion of an object falling.

The straightforward integration method was introduced in Section 1.2.

We need to use a different method of solution for the general equation because the direct method of solution cannot be used.

This method will be used first to make the initial presentation as simple as possible.

The first step is to add up the numbers.

We can integrate if that is the case.

We can compare the left side of the book.

If we can find a solution of Eq, our search for an integrating factor will be successful.

You can easily identify a function that is in line with Eq.

We are going back to Eq.

The left side of Eq.

was chosen by the choice we made of the integrating factor.

Integrating both sides of the book.

Finally, on the problem.

The same solution was found earlier.

The method of integrating factors can be applied to other classes of equations.

In three stages, we will do this.

Consider again Eq.

first.

Multiplying Eq.

Integrating both sides of the book.

By figuring out Eq.

Multiplying Eq.

Integrating both sides of the book.

The axis was requested.

The examples 2 and 3 are special.

The equations considered in Sections 1.1 and 1.2 do not have an equilibrium solution.

If we add up the numbers.

Returning to the beginning.

To find the solution, observe that.

Rewriting something.

To solve the problem.

On the number of Eq.

0 isn't part of the solution of the initial value problem.

There are solutions that behave in two different ways.

If you can solve your equation, you can confirm that the solutions have the property.

The solution obtained in this way agrees with Eq.

The method is discussed in detail in Section 3.7 in relation to second order linear equations.

This process will be shown to be applicable to a much larger class of equations.

You should not get used to using a single pair of letters for the variables in a differential equation.

Linear equations were not considered in the preceding section.

A direct integration process can be used for the subclass of first order equations.

We rewrite Eq.

to identify this class of equations.

There may be other ways as well.

The differential form tends to diminish the distinction between dependent and independent variables.

2 can be separable if you find an equation for its integral curves.

If we write something.

The first term is in Eq.

The same procedure can be followed for any separable equation.

Returning to the beginning.

By combining Eq.

The implicit representation of the solution of the differential equation is called Equation (16).

The determination of an explicit formula for the solution requires that.

We need to solve Eq.

to get the solution.
This is a simple matter.

Only one of the two solutions of the differential equation is satisfactory to the given initial condition.

This is the solution for the minus sign.

The plus sign can be chosen by mistake.

To determine the interval in which the solution is valid, we must find the interval in which the quantity under the radical is positive.

There is a solution to the differential equation.

You can see this by looking at the left side of Eq.

It may seem unlikely that a problem will be simplified by replacing a single equation with a pair of equations but, in fact, the system is more likely to be investigated than the single equation.

If it is convenient to do so, the terminology "solve the following differential equation" means to find an implicit formula for the solution.

Try to come up with an opinion on the advantages and disadvantages of each approach.

Determine where the solution attains its minimum value by solving the initial value problem.

Determine where the solution attains its maximum value by solving the initial value problem.

Determine where the solution attains its maximum value by solving the initial value problem.

Determine where the solution attains its minimum value by solving the initial value problem.

You can use these equations to make predictions about how the natural process will behave.

It is easy to change parameters in the mathematical model over wide ranges, but it can be very time consuming and expensive in an experimental setting.
Both mathematical modeling and experiment or observation have roles in scientific investigations.

On the other hand, mathematical analyses may suggest the most promising directions to explore, and may indicate what experimental data will be most helpful.

We investigated a few simple mathemati cal models in Sections 1.1 and 1.2.

We expand on some of the conclusions reached in those sections.
There are three steps that are always present in the process of mathematical modeling regardless of the field of application.

The steps listed at the end of Section 1.1 are used to translate the physical situation into mathematical terms.

It is critical to state clearly the physical principle that is believed to govern the process.
It has been observed that if the temperature difference is proportional to the rate of growth of the insect population, then objects will move in accordance with the laws of motion.

Each of these statements involves a rate of change that leads to a differential equation.

A mathematical model of the process is the differential equation.

The mathematical equations are only an approximation of the actual process.

For example, bodies moving at speeds comparable to the speed of light are not governed byNewton's laws, insect populations do not grow indefinitely as stated because of eventual limitations on their food supply, and heat transfer is affected by factors other than the temperature difference.

The point of view is that the mathematical equations describe the operation of a simplified physical model so as to embody the most important features of the actual process.

Sometimes the process of mathematical modeling involves the replacement of a single process with a continuous one.
If the population is large, it seems reasonable to consider it a continuous variable and even to speak of its derivative.

Once the problem has been formulated, one is often faced with the problem of solving one or more differential equations, failing that, of finding out as much as possible about the properties of the solution.

If the problem is difficult, further approximations may be indicated at this stage to make the problem manageable.

A linear equation may be approximated by a constant, or a slowly varying coefficients may be replaced by a constant.
To make sure that the simplified mathematical problem still reflects the essential features of the physical process under investigation, any approximations must be examined from the physical point of view.
An intimate knowledge of the physics of the problem can suggest reasonable mathematical approximations that will make the mathematical problem more accessible to analysis.
This interplay of understanding of physical phenomena and knowledge of mathematical techniques and their limitations is characteristic of applied mathematics at its best, and is indispensable in successfully constructing useful mathematical models of intricate physical processes.

After obtaining the solution, you have to interpret it in the context in which the problem arose.

You should always check that the mathematical solution is reasonable.
If possible, calculate the values of the solution at selected points.
Ask if the behavior of the solution is consistent with observations.
Look at the solutions that correspond to the special values of parameters.
The mathematical solution appears to be reasonable, but that doesn't mean it's correct.
If the predictions of the mathematical model are seriously inconsistent with observations of the physical system it purports to describe, this suggests that either errors have been made in solving the mathematical problem, or the mathematical model itself needs refinement, or observations must be made with greater care.

The examples in this section are examples of first order differential equations.

The initial value problem describes the flow process.

We assume that salt is not destroyed in the tank.

The amount of salt varies due to the flows in and out of the tank.

The mixture originally in the tank will eventually be replaced by the mixture flowing in, which has a concentration of 1 lbs/Gal.

The same conclusion can be reached by drawing a direction field.

To solve the problem, note that Eq.

The validity of the model is not in question since this is a hypothetical example.

The differential equation (1) is an accurate description of the flow process if the flow rates are stated and the concentration of salt in the tank is uniform.
It is important that models of this kind are often used in problems involving a pollutant in a lake, or a drug in an organ of the body, rather than a tank of salt water.
It is not easy to determine the flow rates in such cases.
The concentration may be different in some cases.
The variation of the amount of liquid in the problem must also be taken into account because the rates of inflow and outflow may be different.

The differential equation is both linear and separable, which makes it easy to solve the initial value problem.

Consequently, by figuring out Eqs.

A bank account with continuously compounding interest grows larger by the day.

The situation in which compounding occurs at finite time intervals is compared with the results from this continuous model.

The same model applies to more general investments in which dividends and capital gains can accumulate, as well as interest.

The second and third columns are calculated.

The results show that compounding is not important in most cases.

The difference between quarterly and continuous compounding is less than $2 per year during a 10-year period.
For higher rates of return, the difference would be greater than for lower rates.

Let us assume that there may be deposits or withdrawals in addition to the accrual of interest, dividends, or capital gains.

Suppose that one opens an individual retirement account at age 25 and makes annual investments of $2000 in a continuous manner.

The total amount invested is $80,000, so the remaining amount of $500,000 is from the accumulated return on the investment.

The assumed rate is sensitive to the balance after 40 years.

We can now look at the assumptions that went into the model.

We assumed that the return is compounded continuously and that additional capital is invested continuously.
In an actual financial situation, neither of these are true.
We can use expression 16 to determine the approximate effect of different rate projections.

The initial value problem can be used to analyze a number of other financial situations, including annuities, mortgages, and automobile loans.

Consider a pond with 10 million gal of fresh water.

The amount of water in the pond remains constant since the incoming and outgoing flows of water are the same.

The same inflow/outflow principle applies to this example.

If this substitution is made in Eq.

Multiplying Eq.

The equilibrium solution of that equation would be 20.

The mathematical model is adequate for this problem.

There are several assumptions that have not been stated explicitly.

In the first place, the amount of water in the pond is controlled by the rates of flow in and out of the pond.
The chemical flows in and out of the pond, but it is not absorbed by fish or other organisms in the pond.
We assume that the concentration of chemical in the pond is uniform.
The validity of the assumptions used to make the model is very important in determining whether the results are accurate.

The plus sign is used if the body is rising and the minus sign if it is falling back to earth.

There are 20 solutions to the Eq.

The effect of air resistance is not taken into account in the calculation of the escape velocity.

If the body is transported a long distance above sea level before being launched, the effective escape velocity can be reduced.

Air resistance decreases rapidly with increasing altitude.

You should keep in mind that it may be impractical to give space vehicles a large initial speed in a short period of time.

Consider a tank used in an experiment.

A tank has 120 liters of water.

100 gal of fresh water was originally in the tank.

A 500 gal tank originally contained 200 gal of water and 100 lbs of salt.

A tank has 100 gallons of water and 50 ounces of salt.

A college graduate borrows money to buy a car.

A home buyer can not spend more than $800 a month on mortgage payments.

Radiocarbon dating is used in archeological research.

Determine the age of the remains.

The temperature of an object can change at a rate of 19 degrees.

A room with 1200 ft3 of air is free of carbon monoxide.

The information in the table was taken from that source.

A ball with mass 0.15 kg is thrown upward from the roof of a building.

Find the speed of the sky diver when the parachute opens, as a sky diver weighing 180 lbs falls vertically downward.

Determine how long the sky diver is in the air.

A body is falling.

One of the foremost applied mathematicians of the 19th century was George Gabriel Stokes, a professor at Cambridge.

The basic equations of fluid mechanics are named after him, and one of the fundamental theorems of vector calculus is named after him.
He was one of the pioneers in the use of asymptotic series.

We have been primarily concerned with showing that first order differential equations can be used to investigate many different kinds of problems in the natural sciences, and with presenting methods of solving such equations if they are either linear or separable.

It is time to look at some more general questions about differential equations and to look at some important ways in which linear equations differ.

We have encountered a number of initial value problems, each of which had a solution and only one solution.

This raises the question of whether this is true for all first order equations.

This may be an important question for nonmathematicians.

If you encounter an initial value problem in the course of investigating a physical problem, you might want to know that it has a solution before you spend a lot of time and effort trying to find it.
If you are successful in finding a solution, you might be interested in knowing if you should continue looking for other solutions or if you can be sure that there are no other solutions.
The answers to the questions are given by the fundamental theorem.

At a glance, such points can be identified.

The discussion in Section 2.1 leads to the formula.

We can conclude that the differential equation must have a solution by looking more closely at that derivation.

When you add up Eq.

The integrating factor is given by Eq.

We must replace the Theorem 2.4.1 with a more general one.

If the differential equation is linear, the hypotheses in Theorem 2.4.2 reduce to those in Theorem 2.4.1.

The proof of Theorem 2.4.1 was easy to prove because of the expression (3) that gives the solution of an arbitrary linear equation.

The proof of Theorem 2.4.2 is more difficult because there is no corresponding expression for the solution of the differential equation.

In more advanced books on differential equations, it is discussed in greater depth.

Some examples are being considered.

Rewriting something.

You can easily verify that the solution is the same as before.

The differential equation is not applicable to this problem because of the Theorem 2.4.1.

Theorem 2.4.2 doesn't say anything about possible solutions of the problem.

If you want to solve the problem, apply Theorem 2.4.2 to it.

Theorem 2.4.2 does not apply to this problem and no conclusion can be drawn from it.

Since the differential equation is separable, it is easy to solve the problem.

Even at points of discontinuity of the coefficients, solutions can sometimes remain continuous.

The interval in which a solution exists may be difficult to determine for a nonlinear initial value problem.

The following example shows this.

1 is remarkable.

The concept of a general solution is one of the ways in which linear and nonlinear equations differ.

For a first order linear equation it is possible to get a solution containing one arbitrary constant, from which all possible solutions follow by specifying values for this constant.
Even though a solution containing an arbitrary constant may be found, there may be other solutions that cannot be obtained by giving values to this constant.

When discussing linear equations, we will only use the term "general solution" because of the existence of additional solutions.

The situation for equations that are not linear is not as good.

This can only be done for certain types of differential equations, which are the most important.

The equation (26) is an integral, or first integral, of the differential equation, and its graph is an integral curve, or perhaps a family of integral curves.

If you can, arrange for a computer to do this for you.

Examples 1 and 3 in Section 2.2 are examples in which it is better to leave the solution in implicit form and use numerical means to evaluate it for particular values of the independent variable.

It is impossible to find an implicit expression for the solution of a first order equation.

The methods that yield approximate solutions or other qualitative information about solutions are more important because of the difficulty in obtaining exact analytic solutions.

Section 1.1 describes how the direction field of a differential equation can be constructed.
There are interesting features that merit more detailed analytical or numerical investigation.
Section 2.5 talks about graphical methods for first order equations.
There is an introduction to numerical methods for first order equations in Section 2.7 and a discussion of numerical methods in Chapter 8.
It is not necessary to study the numerical algorithms in order to use one of the many software packages that generate and plot numerical approximations to solutions of initial value problems.

There is a general solution that includes all solutions of the differential equation if the coefficients are continuous.

Selecting the correct value for the arbitrary constant can be used to pick out a solution that complies with a given initial condition.

There is an expression for the solution.

If you find the points of discontinuity of the coefficients, you can find the solution.

None of these statements are true.

A solution involving an arbitrary constant may be the only solution for a nonlinear equation.
There is no formula for the solutions of equations.

You are likely to get an equation defining solutions if you integrate a nonlinear equation.

The singularities of solutions of equations can only be found by examining them.
The singularities are likely to be dependent on the initial condition as well as the differential equation.

The equation of part (b) and part (a) are both linear.

Section 3.6 states that solutions of higher order linear equations have a similar pattern.

Sometimes a change of the dependent variable can be used to solve a linear equation.

The method of solution was found by Leibniz.

The independent variable does not appear explicitly in some first order equations.

The growth or decline of the population of a given species is an important issue in fields ranging from medicine to ecology to global economics.

We have already considered the special case in Sections 1.1 and 1.2.

The discussion in Section 2.2 is applicable to Equation (1) because it is separable, but our main object is to show how geometric methods can be used to obtain important qualitative information directly from the differential equation.

The concepts of stability and instability of solutions of differential equations are fundamental to this effort.
Chapter 1 will be examined in greater depth and in a more general setting in Chapter 9.

The population is growing.

There is a problem Solving Eq.

Limitations on space, food supply, or other resources will reduce the growth rate and bring an end to unfettered exponential growth, as it is clear that such ideal conditions cannot continue indefinitely.

The British economist Thomas Malthus first observed that many biological populations increase at a rate that is proportional to the population.

His first paper was on populations.

This function is being used in Eq.

We first look for solutions.

F. Verhulst was a Belgian mathematician.

Logistic growth is what he referred to it as.
He wasn't able to test his model because of insufficient census data, and it didn't receive much attention until many years later.

The direction field has the same slope at all points on each horizontal line, but the slope can change from one line to another.

It seems that 3 is of particular importance.

To understand the nature of the solutions.

These observations show the graphs of solutions.

The fundamental existence and uniqueness of theo rem guarantees that two different solutions never pass through the same point.

If we want to know the value of the population at a particular time, then we need to solve Eq.

The absolute value bars can be removed in this case.

Geometric reasoning can be used to confirm the qualitative conclusions we reached earlier.

A critical point is an unstable equilibrium.

The book by Clark is a good source of information on the population dynamics and economics involved in making efficient use of a renewable resource.

The parameters used here were obtained from a study done by M. S. Mohring on page 53 of the book.

We will see the solutions of Eq.

We conclude that graphs of solutions of Eq.

are the result of all of the information that we have obtained from Figure 2.5.6.

We can confirm the conclusions that have been reached by using the differential equation.

Just as we did for Eq., we separated the variables and integrated them.
However, if we note that.

The threshold phenomenon can be seen in the populations of some species.

The population becomes extinct if too few are present.
Further growth occurs if a larger population can be brought together.
The population can't become unbounded.

Other circumstances also have critical thresholds.

The experimenters want to study the flow of airfoils in a wind tunnel so that the disturbance level is kept low.

The same situation can happen with automatic control devices.

In the normal motion of the plane the changing aerodynamic forces on the flap will cause it to move from its set position, but then the automatic control will come into action to damp out the small deviation and return the flap to its desired position.

The population of the passenger pigeon was present in the United States in large numbers until late in the 19th century.

The population of individual birds declined to extinction in the late 1800s because there wasn't enough of a single place to breed them.

The last survivor died in 1914.
The decline in the passenger pigeon population from huge numbers to extinction in less than three decades was one of the early factors contributing to a concern for conservativism in this country.

A renewable resource is being Harvested.

In the first two chapters of the book, Clark mentioned an excellent treatment of this kind of problem, which goes far beyond what is outlined here.

There are many additional references given there.

The use of mathematical methods to study the spread of diseases goes back to at least one work by Daniel Bernoulli.

Many mathematical models have been proposed and studied for many different diseases.9 Problems 22 through 24 deal with a few of the simpler models and the conclusions that can be drawn from them.
The spread of rumors and consumer products have been described with similar models.

The 20-year-olds have not had the disease.

Some observable quantity, such as a chemical reaction, depends on the physical state of the object.

When the critical value is reached, the reaction suddenly changes its character.
As the amount of one of the chemicals in a certain mixture is increased, spiral wave patterns of varying color suddenly emerge in an originally quiescent fluid.

All equilibrium solutions should be labeled as stable or unstable.

There are a number of integration methods for first order equations.

Linear equations and separable equations are the most important of these.
There is a class of 10 in fluid mechanics.
L. D. Landau was a Russian physicist who won the 1962 Nobel prize for his contributions to the understanding of liquid helium.
He and E. M. Lifschitz wrote a series of physics textbooks.

Most first order equations cannot be solved in this way, so keep in mind that those first order equations that can be solved by elementary integration methods are special.

The methods for those types of equations are not applicable because the equation is not linear or separable.

In solving an equation.

It may not be possible to do this quickly for more complicated equations.

There is a way to determine whether a given differential equation is exact.

There are two parts to the proof.

The first of Eqs is being integrated.

It is not necessary for the region to be rectangular.

The region has no holes in its interior in two dimensions.
For example, a rectangular or circular region is connected, but an annular region is not.
More information can be found in the books.

It is better to go through this process each time, rather than trying to remember the result.

It may or may not be possible to find the solution explicitly, as the solution is obtained in implicit form.

Since the solution of the differential equation is satisfactory, we don't need the most general one.

The first of Eqs is being integrated.

To try to get the second of the Eqs.

There are factors that are integrating.

Sometimes it is possible to convert a differential equation that is not exactly 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- In Section 2.1, we used this procedure to solve linear equations.

The solution found in this way is satisfactory.

A partial differential equation of the form (25) may have more than one solution; if this is the case, any such solution may be used as an integrating factor of Eq.

In practice, integrating factors can only be found in special cases.

A procedure can be used to determine a condition.

Multiplying Eq.

Solutions can be found in explicit form.

You can verify that a second integrating factor is present.

Find the solution if it's exact.

Then solve the equations.

The main exceptions to this statement are differential equations that are linear, separable, or exact, or that can be transformed into one of these types.

The vast majority of first order initial value problems can't be solved by analytical means.

It's important to be able to approach the problem in other ways.

One of the ways to draw a direction field for the differential equation is to visualize the behavior of solutions from the direction field.
Even for complicated differential equations, this is a relatively simple process.
It does not lend itself to quantitative computations or comparisons.

The approximate solution values should be accompanied by error bounds that assure the accuracy of the approximations.

Chapter 8 is devoted to a full discussion of some of the methods that produce numerical approximations to solutions of differential equations.
The oldest and simplest method was invented in 1768.

We can try to do the same thing again.

To use the method, you have to evaluate it.

To find approximate values of the solution.

The values of the actual solution of the initial value problem should be compared with them.

We find the solution in Section 2.1.

The table shows the values of the sand that are different from the values of the computed values.

Some software packages have code for the Euler method, while others don't.

It is easy to write a computer program to carry out the calculations required to produce the results.
The instructions for the program can be written in any high-level programming language.

The numbers can be listed on the screen or printed on a printer, as in the third column of Table 2.7.1.

The results can be seen in graphical form.

The values of the exact solution are presented in Table 2.7.2.

More digits were retained in the intermediate calculations, but the entries are rounded to four decimal places.

It is encouraging that the data confirm our expectations, even though this is what we would expect.

It's too large by only 0.2.
Reducing the step size by a factor of 10 reduces the error by a factor of 10.
An examination of data at intermediate points not recorded in Table 2.7.2 would show where the maximum error occurs for a given step size and how large it is.

The method used by Euler seems to work well for this problem.

Let's look at another example.

The results should be compared with the values of the solution.

For larger step sizes, 5 is 9.4%.

The accuracy that is needed depends on the purpose for which the results are intended, but the errors in Table 2.7.3 are too large for most scientific or engineering applications.
To improve the situation, one can either try smaller step sizes or restrict the computations to a short interval away from the initial point.

It is at each step.

The question of whether the results are accurate enough to be useful must always be kept in mind when using a numerical procedure.

The accuracy of the numerical results could be determined by a comparison with the solution obtained.

The analysis of errors is presented in Chapter 8 and we discuss several more efficient methods than the Euler method.

From a numerical approximation, the best we can hope for is that it reflects the behavior of the actual solution.
A member of a diverging family of solutions will be harder to approximate than a member of a converging family.
Drawing a direction field is a helpful first step in understanding differential equations.

Depending on the type of computing equipment you have, the amount of computing that you can do is reasonable.

Almost any pocket calculator can be used to carry out a few steps of the requested calculations.
For some problems a computer may be needed, while at least a calculator is desirable.

The numerical results may vary depending on how your program is constructed, how your computer executes steps, and so on.

Minor variations in the last decimal place are not necessarily indicative of something being wrong.
The answers in the back of the book are recorded to six digits in most cases.

The results should be compared with those found.

The results should be compared with those found in (a) and (b).

This might cause a difference in the values.

Draw a direction field for the differential equation.

The proof of Theorem 2.4.2 is discussed in this section.

If the differential equation is linear, the existence of a solution of the initial value problem can be established by actually solving the problem and showing a formula for the solution.

This approach is not feasible because there is no way to solve the differential equation in all cases.
For the general case, it is necessary to adopt an indirect approach that shows the existence of a solution.
The core of this method is the construction of a sequence of functions that converges to a limit function satisfying the initial value problem.
The limit function can only be determined in rare cases because it is impossible to compute explicitly more than a few members of the sequence.

The argument is fairly complex and depends on techniques and results that are new to the course.

We don't go into all of the details of the proof, but we point out some of the difficulties involved.

It is necessary to transform the initial value problem into a more convenient form to prove this.

This equation is not a formula for the solution of the initial value problem, but it does provide another relation satisfied by any solution of Eqs.

Any solution of the initial value problem and the integral equation is a solution of the other.

The same conclusion will be used for the initial value problem.

Henri Poincare', perhaps the most distinguished French mathematician of his generation, was appointed professor at the Sorbonne before the age of 30.

He is known for his work in differential equations.
Liouville published a case of the method of successive approximations.
The method is usually credited to Picard, who established it in a general and widely applicable form in a series of papers beginning in 1890.

This doesn't happen, and it's necessary to consider the entire infinite sequence.

We show how these questions can be answered in a relatively simple example, and then discuss some of the difficulties that may be encountered in the general case.

This result can be established with the help of mathematics.

See Eq.

It follows from the beginning.

This isn't necessary for the discussion of existence and uniqueness.

A good approximation to the solution of the initial value problem can be found in this figure.

Multiplying Eq.

Then, after integrating Eq.

Each member of the sequence could be calculated.

As a rule, the members of the sequence cannot be determined.

We use the fact that a continuous function is in a closed region to find such an interval.

There are regions in which successive changes lie.

A sequence of continuous functions can converge to a limit function that is not continuous.

Let's go back to Eq.

We want to interchange the operations of taking the limit on the right side of Eq.

The argument is the same as the one given in the example, and we conclude that the method of successive approximations is the only solution to the initial value problem.

A sequence of continuous functions may converge to a limit function that is not continuous.

This is the same thing.

A continuous model that leads to a differential equation is attractive for many problems, but it may be more natural in some cases.

The continuous model of compound interest used in Section 2.3 is only an approximation to the actual process.

Sometimes population growth can be described more accurately by a discrete model.
This is true for species whose generations do not overlap and thatPropagation occurs at certain times of the year.

The procedure is called iterating the difference equation.

They are important in the study of differential equations.

We will modify the population model.

Natural reproduction and immigration are the sum of those.

We can solve the problem.

By changing the text.

It follows from the beginning.

To deal with that case, we must return to the beginning.

The same model can be used to solve many problems of a financial character.

This is a typical example.

A recent college graduate takes out a loan to buy a car.

The difference equation is Eq.

There is a monthly interest rate of 1%.

000 is given by the author.

Repayment of the principal is $10,000 and interest is $2640.32

Linear equations are easy to understand and have simple solutions.

To make it simpler.

We start our investigation.

One way to look at the question is to approximate it.

Although our argument is not complete, this conclusion is correct.

Substitute from Eq.

Referring to something.

There are graphs of solutions in Figure 2.9.1 The graphs for different initial conditions are the same.

A stairstep diagram is a piecewise linear graph consisting of successive vertical and horizontal line segments.

The process is repeated over and over again.

Heavy portions of the curves show the intervals in which each one is stable.

Neither of the equilibrium solutions is stable.

The values at which the successive period doublings occur approach a limit.

The fine structure of it is unpredictable.
This situation is described by the term.
Extreme sensitivity to the initial conditions is one of the features of chaotic solutions.

Their graphs are quite different after that, as they wander about in the same set of values.

In the last few years, chaotic solutions of difference and differential equations have become widely known.

One of the first instances of mathematical chaos to be studied in detail was Equation 20.
Many questions remain unanswered after the occurrence of chaotic solutions in simple problems.
Chaos may be a part of the investigation of a wide range of phenomena.

Find the effective annual yield of a bank account that pays interest.

An investor deposits money into an account and pays interest at 8% compounded 9.
A college graduate borrows money to buy a car.

A person wants to take out a $100,000 mortgage for 30 years.

A home buyer takes out a $100,000 mortgage with an interest rate of 9%.

A buyer wants to finance the purchase with a $95,000 mortgage with a 20-year term.

The long-term behavior of the solution is independent of the initial conditions.

One of the challenges in solving first order equations is that there are several methods of solution, each of which can be used on a certain type of equation.

It will take some time to become proficient in matching solution methods with equations.
The following problems will help you identify the method or methods of a given equation.

Find the solution that will satisfy the initial condition.

Many more advanced books on differential equations contain a full discussion of the proof of the fundamental existence and uniqueness theorem.