Chapter 6

There are many practical engineering problems that involve mechanical or electrical systems.

The methods described in Chapter 3 are not easy to use.
The Laplace transform is a method that is well suited to these problems.
We describe how this important method works in this chapter, emphasizing problems typical of engineering applications.

In this chapter, we only consider the Laplace1 transform, which is one of the many integral transforms that are useful in applied mathematics.

The transform is defined in this way.

The Laplace transform is useful for equations with constant coefficients that are based on the exponential function.

The Laplace transform is defined by an integral over the range from zero to infinite.

The integral can either fail to exist or fail to exist at all.

Both possibilities are shown in the following examples.

P. S. Laplace studied the relation in 1782.

The techniques described in this chapter were not developed until a century or more later.
Oliver Heaviside, an innovative but unconventional English electrical engineer, made significant contributions to the development and application of electromagnetic theory.

The proof of the result from the math will not be given here.

It is necessary to show that the integral is in Eq.

Theorem 6.1.2 was established.

In this chapter, we only deal with functions satisfying the conditions of Theorem 6.1.2.

In the following examples, the Laplace transforms of some important elementary functions are given.

We use this property frequently and it is of paramount importance.

The Laplace transform can be used to solve initial value problems for linear differential equations.

The relationship is expressed in a formula.

The result is given.

The Laplace transform can be used to solve initial value problems.

It's most useful for problems with nonhomogeneous differential equations.

We begin by looking at some simple equations.

The methods of Section 3.1 can easily be used to solve this problem.

The Laplace transform can be used to solve the problem.

The easiest way to do this is to expand the right side of Eq.

The numerators of the second and fourth members were compared.

The transform of Eq.

can be taken by 2.

By figuring out Eq.

We can point out some of the essential features of the transform method at this early stage.

Laplace transforms can be used to solve linear, constant coefficients, ordinary differential equations if the problem is reduced from a differential equation to an algebraic one.
The task of determining appropriate values for the arbitrary constants in the general solution does not arise if the solution satisfying the initial conditions is found.
Further, as indicated in Eq.

There is a formula for the inverse Laplace transform, but it requires a knowledge of the theory of functions of a complex variable, and we don't consider it in this book.

It is possible to develop many important properties of the Laplace transform without using complex variables.

We did not consider the question of whether there may be functions other than the one given by Eq in the initial value problem.

Functions and Laplace transform in a one-to-one correspondence.
The fact suggests that the transforms of functions frequently encountered can be given by the compilation of a table, such as Table 6.2.1.
The transforms of those in the first column are in the second column.
The inverse transforms of functions in the first and second columns are important.
If the transform of the solution of a differential equation is known, the solution can be found by looking up in the table.
Some of the entries in Table 6.2.1 have been used as examples, while others will be developed later in the chapter.
The third column shows where the derivation of the given transforms can be found.
Table 6.2.1 is sufficient for the examples and problems in this book, but larger tables are also available.
A computer algebra system can be used to get transforms and inverse transforms.

It's convenient to use this property to make use of a given transform by decomposing it into a sum of functions whose inverse transforms can be found in the table.

The useful properties of Laplace transforms are derived later in the chapter.

The Laplace transform and partial fraction expansions are used to solve initial value problems.

The numerator should be expanded on the right side.

The coefficients of the terms on each side are compared.

The governing equations for the Laplace transform were derived in Section 3.8, which is the most important elementary application of the Laplace transform.

Section 3.8 has been noted previously.

There are other physical problems that lead to the differential equation.
Once the mathematical problem is solved, the solution can be interpreted in terms of the physical problem of immediate interest.

There are many initial value problems for second order linear differential equations with constant coefficients in the problem lists.

Many can be seen as models of physical systems, but we usually don't point this out.

The Laplace transforms can be found in the Taylor series expansions.

Problems 28 through 36 are about the Laplace transform.

We show how a general partial fraction expansion can be used to calculate inverse Laplace transforms.

One way to do this is to add up the numbers.

Section 6.2 outlines the general procedure involved in solving initial value problems by means of the Laplace transform.

There are additional properties of the Laplace transform that are useful in the solution of such problems.

The step can be negative.

A rectangular pulse is what this function can be thought of as.

A translation of a function.

One of the useful properties of Laplace transforms is found in the following theorem.

The inverse transform of Eq.

is followed by Equation 6.

The evaluation of certain inverse transforms is the main application of the Theorem.

The results of this section can be used to solve differential equations.

There are examples in the next section.

The nonhomogeneous term, or forcing function, is discontinuous in some examples.

This can be confirmed with a solution to Eq.

The differential equation subject to the initial conditions is an expression for this portion of the solution.

The differential equation becomes Eq.

Laplace transform methods give a more convenient and elegant approach to this problem.

This can also be seen at the same time.

One can show it by direct computation.

The solution itself and its lower derivatives are continuous even at the same points in the differential equation.

The general form of the solution is easy to identify.

4 is a particular solution of the nonhomogeneous equation while the other two terms are the general solution of the corresponding equation.

The solution is about a linear function.

In an engineering context, we might be interested in knowing the amplitude of the steady oscillation.

We indicated earlier that it has a qualitative form.

Explain how the graphs of the solution and forcing function are related.

We observed that an undamped harmonic oscillator 19 in Section 3.9.

It is necessary to deal with phenomena of an impulsive nature in some applications.

According to Eq.

There is no ordinary function of the kind studied in elementary math.

The Laplace transform of the delta function can be formally defined.

To evaluate the limit.

L'Hospital's rule can evaluate its limit.

Paul A. M. Dirac was a professor of mathematics at Cambridge until 1969.

He received the prize for his work in quantum mechanics.
The relativistic equation for the electron was published in 1928.
He predicted the existence of an anti-electron from this equation.
After retiring from Cambridge, Dirac moved to the United States and held a research professorship at Florida State University.

It follows from Eq.

When working with impulse problems, it is convenient to introduce the delta function, and to operate formally on it as if it were a function of the ordinary kind.

This is shown in an example.
It is important to realize that the ultimate justification of such procedures is dependent on a careful analysis of the limiting operations involved.
We don't discuss the mathematical theory here.

The forcing term is the only difference.

There is a decaying oscillation that continues indefinitely.

The second derivative has an infinite discontinuity.

The differential equation requires this since a singularity on one side of the equation must be balanced by a corresponding one on the other side.

This problem is the same as Problem 18.

This problem is the same as Problem 21.

The Laplace transform cannot be commuted with ordinary multiplication.

The situation changes if an appropriately defined "generalized product" is introduced.

Let's make some observations before we give the proof.

The product of the separate transforms is used to give the transform of the two functions.

The reader is responsible for the proof of these properties.

The convolution integral does not have any of the other properties of multiplication.

Systems of this kind are sometimes called hereditary systems and occur in diverse fields such as viscoelasticity and population dynamics.

New variables of integration can make this expression more convenient.

This is the final proof of the Theorem.

The power of the convolution integral as a tool for writing the solution of an initial value problem is shown in example 2.

It is possible to do the same thing in more general problems.

The structure of the solution of this type is given some important insights by the transform approach.

The initial value problem is often referred to as an input-output problem.

By taking the Laplace transform.

The commutative, distributive, and associative properties of the convolution integral are established.

The curve down which a particle will slide freely under gravity, reaching the bottom in the same time regardless of its starting point on the curve.

The clock pendulum has a period that is independent of the motion.
The tautochrone was discovered in 1673 by Christian Huygens using geometric methods.
One of the first times a differential equation was solved explicitly was Bernoulli's solution.

See (b) Combining Eqs.

The tautochrone is an example of a cycloid.

There is a table of transforms in each of the books mentioned.

For example, there are extensive tables available.