Chapter 3

The study of differential equations is dependent on linear equations of second order.

Linear equations have a rich theoretical structure that underlies a number of systematic methods of solution.
A large portion of this structure and methods are understandable at an elementary level.
In order to present the key ideas in the simplest possible context, we describe them in this chapter.
Second order linear equations are vital to any serious investigation of the classical areas of mathematical physics.
It is necessary to solve second order linear differential equations in the development of fluid mechanics, heat conduction, wave motion, or electromagnetic phenomena.
At the end of the chapter, we discuss some basic mechanical and electrical systems.

We can divide Eq.

In talking about Eq.

Chapters 8 and 9 discuss numerical approaches that are more appropriate.
There are two types of second order equations that can be solved by changing variables.

A differential equation is used in an initial value problem.

The values for these two constants are likely to be determined by two initial conditions.

Chapter 4 contains a treatment of higher order linear equations.

Chapter 3 and Chapter 4 can be read in parallel.

In Sections 3.6 and 3.7, we will show that it is possible to express the solution in terms of an integral if the homogeneity equation has been solved.

The more fundamental problem of the equation is the one that needs to be solved.

It turns out that is the case.

It is more difficult to solve Eq.
on the other hand.

Before taking up an exam.

In this example, let's summarize what we've done so far.

It is now possible to pick out a member of this infinite family of solutions that also satisfy a set of initial conditions.

Suppose we want the solution of Eq.

We seek the solution that passes through the point (0, 2) and at that point has a slope of -1.

We differentiate Eq.

By figuring out Eqs.

Finally, put these values in the book.

Let us look for exponential solutions of Eq.

based on our experience.

The differential equation is called Equation 16.

The first case is considered, as is the other two cases in Sections 3.4 and 3.5.

We can differentiate the expression in Eq.

to verify that this is true.

On how to solve Eqs.

It is possible to show that all solutions of Eq.

can be shown on the basis of the fundamental theorem cited in the next section.
We call it Eq.
Any possible initial conditions can be satisfied by the proper choice of the constants.

To use the second initial condition, we need to differentiate Eq.

By figuring out Eqs.

The solution is shown in Figure 3.1.2.

Determine the location of the maximum point.

To find the coordinates of the maximum point from the graph, we need to find the point where the solution has a horizontal line.

The solution of the given differential equation behaves in the same way as any other positive initial slope in this example.

A third case that occurs less often is when the solution approaches a constant when one and the other are negative.

The results give us a better idea of the structure of the solutions of all second order linear equations.

It is helpful to introduce a differential operator notation in the theory of linear differential equations.

We would like to know if anything can be said about the form and structure of solutions that might be helpful in finding solutions of particular problems.

This section contains the answers to the questions.

The fundamental theoretical result for initial value problems for second order linear equations is similar to the one for first order linear equations.

The result applies equally well to nonhomogeneous equations.

Some of the assertions are easy to prove.

We found a solution to the initial value problem.

It is difficult to show that the initial value problem has no solutions other than the one given by Eq.
The only solution of the initial value problem is stated in Theorem 3.2.1.

It is not possible to write down a useful expression for most problems of the form.

There is a major difference between first and second order equations.
General methods that don't involve having an expression are the best way to prove parts of the theorem.
We don't discuss the proof of Theorem 3.2.1 here, but we will use it whenever necessary.

The given differential equation can be written in Eq.

This is the longest interval in which the solution is guaranteed.

It is the only solution of the problem that is unique.

A proof of Theorem 3.2.1 can be found in Chapter 6, Section 8 of the book.

Theorem 3.2.2 states that, beginning with two solutions of Eq.

The next question is whether all the solutions are the same.

After solving Eqs.

The following result can be established by the preceding argument.

Jo'sef Maria Hoe"ne'-Wronski was born in Poland but spent most of his life in France.

Wronski's life was marked by frequent heated disputes with other individuals and institutions, and he was a gifted but troubled man.

This is the final proof of the Theorem.

We need only two solutions of the given equation whose Wronskian is nonzero to find the general solution of the equation.

We did this in several examples, but we did not calculate the Wronskians.
You should verify that the general solutions in Section 3.1 do satisfy the Wronskian condition by going back and doing that.
The following example includes all the problems of a similar type, as well as those mentioned in Section 3.1.

Section 5.5 will show how to solve Eq.

The general solution of a given differential equation has been found in several cases.

The question may arise as to whether or not a differential equation of the form always has a fundamental set of solutions.
The affirmative answer to this question is provided by the following theorem.

The difficult part of this proof, demonstrating the existence of a pair of solutions, is taken care of by reference to Theorem 3.2.1.

We noted in Section 3.1 that there were two solutions.

We need to find the solutions that satisfy the proper initial conditions to find the fundamental solutions.

You should choose the set that is most convenient.

The discussion can be summarized as follows.

Explain why this result doesn't conflict with Theorem 3.2.2.

The original equation is Airy's equation of the adjoint equation.

The concept of linear independence of two functions is related to the representation of the general solution of a second order linear homogeneous differential equation as a linear combination of two solutions.

This is a very important idea that has significance far beyond the present context, and we briefly discuss it in this section.

The following is the basic property of systems of linear equations.

It is easy to answer the question of whether a large set of functions is linearly dependent or linearly independent if they are proportional to each other.

These definitions are illustrated by the following examples.

Evaluating the work.

Since the determinant is not zero, the only solution is Eq.

Linear independence and dependence are related to the Wronskian.

The first part of the Theorem is followed by the second.

You should not read too much into it.

Let's take a closer look at the properties of the Wronskian of two solutions of a second order linear differential equation.

The formula for the Wronskian of any two solutions of any equation, even if the solutions themselves are not known, is given by the following theorem.

We can write Eq.

after that.

The result in Theorem 3.3.2 was derived from the work of the Norwegian mathematician.

The question of how to solve a quintic, or fifth degree, polynomial equation has been open since the 16th century.
His greatest contribution was in the study of elliptic functions.
His work was not noticed until after he died.

Since it is a first order linear equation and a separable equation, it can be solved immediately.

The Wronskian of any two sets of solutions of the same differential equation can only be determined by a multiplicative constant, and can't be solved.

The Wronskian of any pair of solutions is given by Equation 15.

If the two functions involved are solutions of a second order linear differential equation, a stronger version of Theorem 3.3.1 can be established.

The facts about fundamental sets of solutions, Wronskians, and linear independence can be summarized in the following way.

There is a similarity between second order linear homogeneous differential equations and two-dimensional vector algebra.

The basis for the two-dimensional vectors is said to be a pair of linearly independent vectors.

A good reason for studying abstract linear algebra is the connection between differential equations and vectors.

2 2 is a linearly independent set of solutions.

The first task is to explore what is meant by these expressions, which involve evaluating the exponential function for a complex exponent.

The answer is provided by a formula.

To assign a meaning to the expressions.

When the exponent is real, we want the definition to be reduced to the familiar real exponential function.
There are many ways to extend the exponential function.

The equation (9) is an important mathematical relationship.

We have put matters on a firm foundation.

There are some variations of the formula.

We are now taking Eq.

The real and imaginary parts of a complex number are given by the terms on the right side of the equation.

It is easy to show that the usual laws of exponents are valid for the complex exponential function with the definitions.

The second initial condition must be differentiated.

We will talk about the geometrical properties of solutions in Section 3.8, so we will be brief here.

Solutions are decaying.
The solution doesn't grow nor decay, but it does oscillate, a typical solution of Eq.

Substitute the expressions in Eqs.

Jean d'Alembert was a French mathematician who was known for his work in mechanics and differential equations.

The wave equation first appeared in his paper on vibrating strings in 1747, and the principle in mechanics and d'Alembert's paradoxes are named for him.
He was the science editor of Diderot's Encyclope'die in his later years.

We assume that the coefficients are in Eq.

Substitute in Eq.

There is a problem that we are considering.

To satisfy the second initial condition, we first differentiate.

When the roots are real and different, the geometrical behavior of solutions is the same as it is in this case.

The procedure used earlier in this section for equations with constant coefficients is more generally applicable.

The variables are separated in Eq.

Plot the solution to the initial value problem.

Consider the initial value problem and solve it.

The value zero can be taken on by 0 at most times.

See the references given there.

The structure of solutions of the nonhomogeneous equation is described in the following two results.

The proof of Theorem 3.6.2 is very similar to the preceding one.

2 2 is the same as Eq.

This solution is often referred to as a specific solution.

The functions are found in the first two steps.

We want to discuss two methods.

The method of variation of parameters is also known as the method of undetermined coefficients.

There are some advantages and some possible drawbacks.

The assumed expression is replaced by Eq.

This means that there is no solution of the form that we assumed if we cannot determine the coefficients.
We can try again if we modify the initial assumption.

It is only useful for equations that we can easily write down the correct form of the solution in advance.

This method is usually only used for problems in which the equation has constant coefficients and the nonhomogeneous term is restricted to a relatively small class of functions.
We only consider nonhomogeneous terms that consist of polynomials, exponential functions, and sines.
The method of undetermined coefficients is useful for many problems that have important applications.
The computer algebra system can be useful in practical applications.
We will show the method of undetermined coefficients by several simple examples and then explain some rules for using it.

Substitute in Eq.

When the right side of the equation is a polynomial, the method shown in the preceding examples can be used.

These expressions can be replaced in Eq.

This procedure is illustrated in the following example.

By dividing the right side of the book.

The procedure illustrated in these examples can be used to solve a lot of problems.

Sometimes there is a difficulty.
The next example shows how it happens.

There is no solution to Eq.

To find a solution.

It will not be necessary to carry the process further than this for a second order equation.

The general solution of the equation can be found.

The method of variation of parameters can be used if this is not the case.

It is necessary to remove the duplication.

The sum of the general solution and the particular solution of the nonhomogeneous equation is needed.

The nonhomogeneous equation has a general solution.

The values of the arbitrary constants remaining in the general solution can be determined using the initial conditions.

It is easy to carry out this procedure by hand, but in many cases it requires a lot of math.

A computer algebra system can help execute the details if you understand how the method works.

If one assumes too many terms, some unnecessary work is done and some coefficients turn out to be zero, but at least the correct answer is obtained.

The method of undetermined coefficients was described in the preceding discussion.

-1 + * * + a.

Substituting in the book.

We can't satisfy Eq.

The latter form is usually preferred.

We describe another method of finding a solution to a nonhomogeneous equation.

This method is similar to the method of undetermined coefficients.
There is a more general discussion of factoring operators.

This method is used to derive a formula for a particular solution of a linear nonhomogeneous differential equation.

The method of variation of parameters eventually requires that we evaluate certain integrals involving the nonhomogeneous term in the differential equation, and this may present difficulties.
This method is used in an example before we look at it in the general case.

We need a different approach.

The equation corresponding to Eq.
is also a homogeneous equation.

It is possible to choose the second condition in a way that makes the computation more efficient.

Now going back to Eq.

It follows from the beginning.

By differentiating Eq.

Solving Eq.

is an example.

Substitute these expressions in Eq.

The method of variation of parameters worked well in determining a particular solution and the general solution of Eq.

This is a major assumption because we have shown how to solve it.

We differentiate Eq.

By combining Eqs.

Substitute from Eq.

2 2 is prescribed by the Theorem 3.6.2.

There may be two major difficulties in using the method of variation of parameters if we look at the expression (28) and the process by which we derived it.

The evaluation of the integrals is a possible difficulty.

It was used in using Eq.

The method of variation of parameters has an advantage.

If you want to investigate the effect of variations in the forcing function, or if you want to analyze the response of a system to a number of different forcing functions, this expression is a good starting point.

You can check your answer by using the method of undetermined coefficients.

The fields of mechanical and electrical oscillations are important areas of application.

Understanding the behavior of a mass on a spring is the first step in the investigation of more complex vibrating systems.

Many problems are similar to the principles involved.

We are interested in studying the motion of the mass when it is displaced or acted on by an external force.

The spring force is directed upward after the spring is extended.

The force that the spring exerts on the mass is always expressed by Eq.

Resistance from the air or other medium in which the mass moves, internal energy dissipation due to the extension or compression of the spring, or a mechanical device are some of the sources of this force.

The mass is usually referred to as scrutineers.

Hooke was an English scientist with many interests.

The assumption that the force is modeled adequately is rather complicated.

Dashpots do behave as Eq.
An important benefit of the assumption is that it leads to a linear differential equation.
This means that a thorough analysis of the system is straightforward, as we will show in this section and the next.

The force could be due to the motion of the mount to which the spring is attached, or it could be applied directly to the mass.

The external force can be periodic.

It's important to understand that.

The mass of the attached body has been neglected in comparison with the mass of the spring.

The conditions give a mathematical problem that has a unique solution.

Our intuition tells us that if the mass is set in motion with a given initial displacement and velocity, then its position will be determined uniquely at all future times.
The solution of Eq gives the position of the mass.

A mass is stretching a spring.

The motion of the mass is governed by the initial value problem.

Our task is to determine the various constants in the differential equation and initial conditions so that we can solve the required initial value problem.

The units of measurement are the first thing to choose.
It is natural to use the English language instead of the metric system of units.
The foot and inch appear in the statement as units of length.
Having made a choice, it is important to be consistent.

The second initial condition is implied by the word "released" in the statement of the problem, which we believe to mean that the mass is set in motion with no initial velocity.

This is an idealized configuration of the system that is not always doable in practice.

It is possible to get satisfactory results over short to moderate time intervals if the assumption of no damping is made.

The solution of Eq.

was discussed.

By comparing the two.

A displaced cosine wave is a description of a periodic, or simple, motion of the mass.

The motion was described by Eq.

There is no way for the system to get rid of the energy that comes from the initial displacement and velocity.

A mass of 10 lbs stretches a spring of 2 in.

Determine the period, amplitude, and phase of the motion.

There are two solutions of the equation, one in the second and the other in the fourth.

Our expectation is that the motion dies out with increasing time because of the gradual dispersal of the energy.

The third one is the most important because it occurs when the damping is small.

The motion is called a damped vibrating motion.

The effect of small damping is to reduce the frequencies of the oscillation.

The time between the position of the mass and its equilibrium position is known as the maxima or minima.

The quasi period is increased by small damping.

As stated by Eqs.

The cases were given by Eqs.

Determine the position of the mass at any time.

We show the motion if the term is neglected.

The critical value is one-sixteenth of the small damping coefficients in this example.

The amplitude of the oscillation is reduced quickly.

The units have been chosen so that 1 ampere is equal to 1 coulomb.

The most important conclusion from this discussion is that the flow of current in the circuit is described by an initial value problem of precisely the same form as the one that describes the motion of a spring-mass system.

If you know how to solve second order linear equations with constant coefficients, you can interpret the results in a variety of ways.

One of the leading physicists of the 19th century was 9Gustav Kirchhoff.

He is famous for his work in emission and absorption, as well as being one of the founding fathers of spectroscopy.

A mass is stretching a spring.

3 in.
is added to the mass if it is pulled down.

A mass is stretching a spring.

The mass is attached to a 10.

A mass is stretching a spring.

A spring is stretched by 3 newtons.

The spring 12 has a mass of 2 kilograms hanging from it.

A mass is stretching a spring.

Find the solution of the initial value problem if the undamped spring-mass system is in the right position.

The principle of conserve energy should be confirmed by your result.

First suppose that there isn't any damping.

There are two interesting cases.

The two periodic functions of different periods are the same.

We are able to write Eq.

There is a phenomenon when two tuning forks are sounded at the same time.

The periodic variation of the sound is obvious to the untrained ear.

In reality, oscillations do not occur.

Depending on the circumstances, resonance can be either good or bad.

It must be taken very seriously in the design of structures, such as buildings or bridges, where instabilities can lead to catastrophic failure.

The space shuttle main engine has a high-pressure fuel turbocharger.

The design speed of 39,000rpm was not enough to operate the turbocharger over 20,000rpm.

In the design of seismographs, resonance can be used to detect weak signals.

There are forced impacts with damp.

The solution allows us to satisfy whatever initial conditions are imposed; with increasing time the energy put into the system by the initial displacement and velocity is dissipated through the damping force, and the motion becomes the response of the system to the external force.

The effect of the initial conditions would persist if there was no damping.

It follows from Eqs.

They rise and fall together, and in particular, assume their respective maxima nearly together and their respective minima nearly together, because the response is nearly in phase with the excitation.

For comparison, the graph of the forcing function is shown.

A mass is stretching a spring.

The mass was displaced by 2 in.

An external force of 7 acted on the mass.

If an undamped spring-mass system with a mass that weighs 6 lbs and a spring constant 10.

A mass is stretching a spring.

A spring is stretched.

The mass is attached to a dashpot.

A spring constant of 3 N/m is what a spring-mass system has.

There is a mass attached to 13 Determine when the steady-state response given by Eq will be.

There are many books on electric circuits.