Chapter 1

• We try to give perspective to your study of differential equations in this chapter.
• We use two problems to illustrate some of the basic ideas that we will return to and elaborate upon frequently throughout the rest of the book.
• In order to provide organizational structure for the book, we indicate several ways of classifying equations.
• Some of the major figures and trends in the historical development of the subject are outlined.
• The study of differential equations has attracted the attention of many of the world's greatest mathematicians.
• There are many interesting open questions in the field of inquiry.

• For some students the interest in the subject itself is enough motivation, but for most it is the chance of important applications to other fields that makes the undertaking worthwhile.

• Many of the principles, or laws, underlying the behavior of the natural world are related to rates at which things happen.
• The relations are equations and the rates are derivatives.

• A math-ematical model of the process is one of the models discussed in the book.
• Two models lead to equations that are easy to solve in this section.
• The simplest differential equations can provide useful models of physical processes.

• An object is falling in the atmosphere.

• The letters represent various quantities of interest in the problem.
• A falling to time.

• The choice of units of measurement is somewhat arbitrary, and there is nothing in the statement of the problem to suggest appropriate units, so we are free to make any choice that seems reasonable.

• Newton's second law states that the mass of the object is equal to the net force on the object.

• It is more difficult to model the force due to air resistance.
• This is not the place for an extended discussion of the drag force; suffice it to say that it is often assumed that the drag is proportional to the velocity, and we will make that assumption here.

• There is a mathematical model of an object falling in the atmosphere.
• It is common to refer to them as parameters, since they may take on a range of values during an experiment.

• There is a diagram of the forces on a falling object.

• To solve the problem.
• We will show you how to do this in the next section.
• Let's see what we can learn about solutions without actually finding them.

• Determine the behavior of solutions.

• We will look at Eq.
• Then, by looking at the right side of the book.

• Referred to Eq.
• again.

• The solution is 49.

• There is a balance between drag and gravity.
• The direction field has 49 superimposed on it.

• Each line segment is related to a graph of the solution passing through it.
• A direction field drawn on a grid gives a good picture of the behavior of solutions of a differential equation.
• The construction of a direction field is the first step in the investigation of a differential equation.

• There are two observations that are worth mentioning.
• In constructing a direction field, we don't have to solve Eq.
• Even equations that may be difficult to solve can be constructed with direction fields.
• You should use a computer to draw a direction field if you want to evaluate a given function multiple times.

• Let's look at another example.
• Consider a group of field mice in a rural area.
• We assume that the mouse population increases at a rate that is proportional to the current population.

• Assume that several owls live in the same neighborhood and that they kill 15 field mice per day.
• A better model of population growth is discussed later in Section 2.5.

• Time is measured in months and the monthly rate is needed, so the term is -450 instead of -15.

• Determine the solutions of Eq.

• The equilibrium solution is very important in understanding how solutions of a differential equation behave.

• The solutions of this equation are very similar to those of Eq.

• The models discussed in this section have limitations.
• As soon as the object hits the ground, the model ceases to be valid.
• This model becomes unacceptable after a short time interval because of the unrealistic predictions.

• Making mathematical models.

• The appropriate differential equation that describes the problem being investigated is the first thing that needs to be applied to any of the numerous fields in which differential equations are useful.
• We looked at two examples of this modeling process, one from physics and the other from ecology.
• Success in modeling is not a skill that can be reduced to obeying a set of rules.
• A satisfactory model is the most difficult part of the problem.

• Determine the independent and dependent variables and assign letters to represent them.
• Time is the independent variable.

• For each variable, choose the units of measurement.
• Some choices may be more convenient than others.
• We chose to measure time in seconds in the falling object problem and months in the population problem.

• Understand the basic principle that governs the problem you are investigating.
• This may be a widely recognized physical law, such as the law of motion, or it may be a more speculative assumption based on your own experience.
• This step is not likely to be a purely mathematical one, but will require you to be familiar with the field in which the problem lies.

• In step 3, express the principle or law in terms of the variables you chose.
• It may require the introduction of physical constants or parameters.
• The use of auxiliary or intermediate variables may be related to the primary variables.

• Each term in your equation should have the same physical units.
• If this isn't the case, you should seek to repair your equation.
• If the units agree, your equation is consistent, although it may have other flaws that this test does not reveal.

• The desired mathematical model can be found in the result of step 4.
• It is important to keep in mind that in more complex problems the resulting mathematical model may be much more complicated.

• A pond contains an unknown amount of water and a differential equation that shows the amount of chemical in it.
• The raindrop volume is a function of time.

• A hospital patient is receiving a drug.
• The drug is always uniformly distributed throughout the bloodstream.

• qualitative conclusions were drawn about the behavior of solutions of Eqs.
• To answer questions of a quantitative nature, we need to find solutions of our own.

• The interaction of certain populations of field mice and owls is described in Field Mice.

• One way to go is here.

• Integrating both sides of Eq.

• Taking the exponential of both sides of Eq.

• 900 is a solution of Eq.

• The 900 solution tends to differ from that solution.

• This is what happens when you solve a differential equation.

• The solution process involves an integration, which brings with it an arbitrary constant, whose possible values generate an infinite family of solutions.

• We want to focus our attention on a single member of the infinite family of solutions by specifying the value of the arbitrary constant.
• We usually do this by specifying a point on the graph of the solution.

• This value can be inserted in Eq.

• Taking the exponential of both sides of the equation.

• There are all possible solutions of Eq.
• in savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay savesay

• Identifying the integral curve that passes through the initial point is what satisfying an initial condition entails.

• To relate the solution to the problem.

• The first thing to do is to state an appropriate initial condition.

• The velocity of the falling object is given by Equation (26) before it hits the ground.

• We need to know when the impact occurs to find the object'svelocity.

• Substitute these values in Eq.

• We have discussed differential equations with mathematical models of a falling object and a hypothetical relation between field mice and owls before.
• The ultimate test of any mathematical model is whether its predictions agree with observations or experimental results, and the derivation of these models may have been plausible, but you should remember that.
• There are several sources of possible discrepancies that we don't have any actual observations or experimental results to compare.

• The underlying physical principle ofNewton's law of motion is applicable in the case of a falling object.
• It is not certain if the drag force is proportional to the speed.

• Various uncertainties affect the model of the mouse population.
• As the population gets smaller, a constant predator rate becomes harder to sustain.
• The model predicts that the population will grow larger and larger.

• If a mathematical model is incomplete or inaccurate, it may never be useful in explaining qualitative features of the problem under investigation.

• It can be valid under certain circumstances but not others.
• Good judgement and common sense should always be used when constructing mathematical models.

• If you modify example 2, the falling object will not experience air resistance.

• The time required for the thorium-234 to decay to one-half its original amount is proportional.

• The half-life of radium-226 is 1620 years.
• Find the time period in which a given amount of material is reduced.

• Your swimming pool contains 60,000 gal of water and it has been contaminated with a dye that leaves a swimmer's skin green.
• The equation for the pool's filtering 3 is derived from the laws of Kirchhoff.

• The main purpose of this book is to discuss some of the properties of solutions of differential equations, and to describe some of the methods that have proved effective in finding solutions, or in some cases approximating them.
• The framework for our presentation is provided by the ways in which differential equations are classified.

• One of the more obvious classifications is based on whether the function depends on a single independent variable or several independent variables.

• Ordinary differential equations are the ones discussed in the preceding two sections.

• There are certain physical constants.
• The wave equation arises in a variety of problems involving wave motion in fluids orsolids, and the heat conduction equation describes the conduction of heat in a solid body.

• differential equations are classified by the number of unknown functions involved.
• One equation is enough if there is a single function to be determined.
• A system of equations is required if there are two or more unknown functions.
• The Lotka-Volterra equations are important in ecological modeling.

• The Lotka-Volterra equations are examined in section 9.5 of the systems of equations.
• It's not unusual to encounter systems with a lot of equations.

• The equations in the preceding sections are first order.
• The equations are partial differential equations.

• We only study the equations of the form.
• A single equation of the form may correspond to several equations of the form.

• The classification of differential equations is important.

• The equations in Sections 1.1 and 1.2 describe the falling object and the field mouse population and are linear.

• A simple physical problem that leads to a differential equation is the pendulum.

• Linear equations are solved using the mathematical theory and methods.
• The methods of solution are not as satisfactory for nonlinear equations.
• It is fortunate that many significant problems lead to linear ordinary differential equations or can be approximated by linear equations.

• The simpler parts of the subject are emphasized in an elementary text.
• Linear equations and various methods for solving them are the focus of the book.
• Chapters 8 and 9, as well as parts of Chapter 2, are concerned with equations that are not straight.
• When it is appropriate, we point out why linear equations are more difficult and why many of the techniques used to solve them can't be applied to nonlinear equations.

• In Section 1.2 we found solutions to certain equations by direct integration.

• It can be difficult to find solutions of differential equations.
• If you find a function that you think may be a solution of a given equation, it is easy to determine if the function is actually a solution by substituting it into the equation.

• There are too many possible functions for you to have a good chance of finding the correct one, so this is not a good way to solve differential equations.
• It is important to know that you can verify the solution is correct by substituting it into the differential equation.
• This can be a very useful check for a problem of any importance, and you should make a habit of considering it.

• We are able to verify that certain simple functions are solutions, but in general we don't have such solutions readily available.
• For at least two reasons, this is not a purely mathematical concern.
• If a problem doesn't have a solution, we would prefer to know that before we try to solve it.
• If a sensible physical problem is modeled as a differential equation, then the equation should have a solution.
• Presumably there is something wrong with the formula if it does not.
• An engineer or scientist checks the validity of a mathematical model.

• If we assume that a differential equation has at least one solution, the question arises as to how many solutions it has, and what additional conditions must be specified to single out a particular solution.
• One or more arbitrary constants of integration can be found in the solutions of differential equations.

• Practical implications of the issue of uniqueness also exist.
• We can be sure that we have solved the problem if we know that the problem has a unique solution.
• Maybe we should continue to search for other solutions.

• We have answered the question of the existence of a solution if we find a solution to the equation.
• Without knowledge of existence theory, we could use a computer to find a numerical approximation to a solution that doesn't exist.

• It is possible that the solution is not expressible in terms of the usual elementary functions.
• This is the situation for most differential equations.
• It is important to consider methods of a more general nature that can be applied to more difficult problems when discussing methods that can be used to get solutions of relatively simple problems.

• The study of differential equations can be done with a computer.
• Computers have been used for many years to construct numerical approximations to solutions of differential equations.
• The current level of generality and efficiency has been refined to an extremely high level.
• A few lines of computer code, written in a high-level programming language and executed on a relatively inexpensive computer, is enough to solve a wide range of differential equations.
• More advanced routines are also available.
• These routines combine the ability to handle large and complicated systems with many diagnostic features that alert the user to possible problems as they are encountered.

• A table of numbers is the usual output from a numerical program, listing the values of the independent and dependent variables.
• It is easy to show the solution of a differential equation graphically, whether the solution has been numerically obtained or the result of an analytical procedure.
• Several well-crafted and relatively inexpensive special-purpose software packages are available for the graphical investigation of differential equations.
• Powerful computation and graphical capability can be found within the reach of individual students thanks to the widespread availability of personal computers.
• In the light of your own circumstances, you should consider how best to use the available computing resources.

• The availability of extremely powerful and general software packages that can perform a wide variety of mathematical operations is very relevant to the study of differential equations.
• There are three packages that can execute extensive numerical computations.
• They can solve many differential equations in response to a single command.
• If you want to deal with differential equations in more than a superficial way, you should be familiar with at least one of these products.

• The computing resources have an effect on how you study differential equations.
• To become confident in using differential equations, it is necessary to understand how the solution methods work, and this understanding is achieved by working out a sufficient number of examples in detail.
• Eventually, you should plan to delegate as much as possible of the routine details to a computer, while you focus more attention on the proper interpretation of the solution.
• The best methods and tools should always be used for each task.
• Combining numerical, graphical, and analytical methods will allow you to understand the behavior of the solution and the underlying process that the problem models.
• Some tasks can be done with pencil and paper, while others require a calculator or computer.
• Good judgement is needed when selecting a combination.

• To derive the equation of motion of a pendulum, follow the steps indicated here.

• Assume that the rod is rigid and weightless, that the mass is a point mass, and that there is no drag in the system.

• The forces are acting on the mass.

• The force in the rod doesn't enter the equation.
• The component of the force needs to be found in the tangential direction.

• It is difficult to appreciate the history of this important branch of mathematics without knowing something about differential equations and methods of solving them.
• The general development of mathematics is intertwined with the development of differential equations.
• There are footnotes throughout the book and references at the end of the chapter.

• The study of differential equations began in the 17th century.
• In 1669, Lucasian Professor of Mathematics at Trinity College, Cambridge, was born, and he grew up in the English countryside.
• The basis for their applications in the eighteenth century was provided by the development of the calculus and the elucidation of the basic principles of mechanics byNewton, who did relatively little work in differential equations as such.
• The revision and publication of results obtained much earlier endedNewton's active research in mathematics in the early 1690s.
• He resigned his professorship a few years after he was appointed the British Mint's warden.
• He was buried in the abbey after he was knighted.

• At the age of 20 he received his doctorate in philosophy from the University of Altdorf.
• He worked in several different fields.
• His interest in mathematics developed when he was in his twenties.
• The first to publish the fundamental results of calculus was Leibniz, who arrived at them independently.
• He discovered the method of separation of variables in 1691, the reduction of equations to separable ones in 1691, and the procedure for solving first order linear equations in 1694.
• He spent his life as an ambassador and adviser to several German royal families, which allowed him to travel widely and carry on an extensive correspondence with other mathematicians.
• Many problems in differential equations were solved during the last part of the 17th century.

• The Bernoulli brothers did a lot to develop methods of solving differential equations.
• After his brother's death in 1705 Johann was appointed to the same position as his brother's professor of mathematics.
• Both men were often involved in disputes with each other.
• Both made significant contributions to mathematics.
• They were able to solve a number of mechanics problems by using differential equations.
• The term "integral" was first used in the 1690 paper.

• The brachistochrone problem was solved by the Bernoulli brothers.
• It is said thatNewton solved the problem after dinner after learning of it late in the day at the Mint.

• Daniel Bernoulli went to St. Petersburg as a young man to join the academy, but returned to his native Switzerland in 1733 as a professor of physics.
• His interests were mostly in partial differential equations.
• His name is associated with the Bernoulli equation in fluid mechanics.
• He was the first to see the functions that became known as Bessel functions.

• The greatest mathematician of the 18th century was a student of Bernoulli.
• He went to St. Petersburg with Daniel Bernoulli.
• He was associated with the Berlin Academy for the rest of his life.
• His works fill more than 70 large volumes, and he was the most prolific mathematician of all time.
• His interests ranged from mathematics to many fields of application.
• Even though he was blind for 17 years, he continued to work until his death.
• The development of methods of solving mathematical problems is of particular interest to him.
• In 1734-35, he identified the condition for exactness of first order differential equations, developed the theory of integrating factors, and gave the general solution of homogeneous linear equations with constant coefficients.

• Power series was used a lot in solving differential equations.
• He made important contributions in partial differential equations and gave the first systematic treatment of the calculus of variations by proposing a numerical procedure.

• At the age of 19, Joseph-Louis Lagrange became a professor of mathematics.
• He moved to the Paris Academy in 1787 after taking over the chair of mathematics at the Berlin Academy in 1766.
• He gave a complete development of the method of variation of parameters.
• In partial differential equations, lagrange is known for its fundamental work.

• After living in Normandy as a boy, Pierre-Simon de Laplace came to Paris in 1768 and was elected to the Acade'mie des Sciences.
• Laplace studied his equation extensively in relation to the attraction of stars.

• The Laplace transform was not used in differential equations until much later.

• By the end of the 18th century there were many elementary methods of solving differential equations.
• The development of less elementary methods such as those based on power series expansions was one of the things that interest turned to in the 19th century.
• The methods find their setting in the plane.
• They were stimulated by the simultaneous development of the theory of complex analytic functions.
• As their role in mathematical physics became clear, partial differential equations began to be studied more and more.
• A number of functions, arising as solutions of ordinary differential equations, occurred repeatedly and were studied thoroughly.
• Bessel, Legendre, Hermite, Chebyshev, and Hankel are some of the mathematicians whose names are associated with higher transcendental functions.

• The investigation of methods of numerical approximation was led by the many differential equations that resisted solution by analytical means.
• The implementation of numerical integration methods was severely restricted by the need to use very primitive computing equipment.
• The range of problems that can be investigated using numerical methods has vastly increased in the last 50 years due to the development of increasingly powerful and versatile computers.
• During the same time period, extremely refined and robust numerical integrators have been developed.

• The ability to solve many significant problems within the reach of individual students has been brought about by versions appropriate for personal computers.

• In the twentieth century, the creation of geometric or topological methods has been a characteristic of differential equations.
• The goal is to understand at least the qualitative behavior of solutions from a point of view.
• More detailed information can be obtained using numerical approximations.
• Chapter 9 contains an introduction to these methods.

• The two trends have come together in the past few years.
• The study of systems of differential equations has been stimulated by computers and computer graphics.
• Unexpected phenomena, referred to by terms such as strange attractors, chaos, and fractals, have been discovered, are being studied, and are leading to important new insights in a variety of applications.
• Differential equations at the dawn of the twenty-first century are a fertile source of fascinating and important unsolved problems.

• The differential equations software in the book changes too fast for particulars to be given.

• There are a number of books that deal with the use of computer systems for differential equations.