Chapter 4

Chapter 4

The theoretical structure and methods of solution for second order linear equations are applicable to third and higher order equations.

In this chapter, we take particular note of those instances where new phenomena may appear, due to the greater variety of situations that can occur for equations of higher order.

There is a mathematical theory associated with Eq.

The proof of most of the results are similar to the proof of the second order equation.

An arbitrary constant is introduced by each of these integrations.

The existence and uniqueness theorem is similar to the one used in Theorem 3.2.1.

We won't give a proof of it here.

We don't know if it is unique without using the Theorem 4.1.1.

The proof of the theorem can be found in Ince.

The following is a theorem.

In the same way as for the second order linear equation, the existence of a fundamental set of solutions can be demonstrated.

Section 3.3 talks about linear dependence and independence.

The difference of two solutions of the nonhomogeneous equation is a solution of the homogeneous equation.

The general solution of the nonhomogeneous equation is called the linear combination.

This is a fairly simple problem if coefficients are constants.

If coefficients are not constants, it is necessary to use numerical methods similar to those in Chapter 5.
As the order of the equation increases, these become more cumbersome.

The reduced equation is at least of second order and rarely will it be simpler than the original equation.

Reduction of order is not useful for equations of higher order.

We show how to generalize Theorem 3.3.2 to higher order equations.

The differential equation has a characteristic equation.

If the functions are linearly independent, the general solution of Eq.

For more than 200 years, an important question in mathematics was whether every equation has at least one root.

The answer to this question was given by Carl Friedrich Gauss in his PhD in 1799, but his proof does not meet modern standards of rigor.
Three of Gauss's proofs have been discovered.
In a first course on complex variables, students often meet the fundamental theorem of algebra, where it can be established as a consequence of some of the basic properties of complex analytic functions.

There are formulas that give exact expressions for the roots for third and fourth degree polynomials.

Calculators and computers can be used to find the root of a problem.

Sometimes they are included in the differential equation solvers so that the process of calculating the characteristic polynomial is hidden and the solution of the differential equation is produced automatically.

One result that is sometimes helpful is if you need to factor the characteristic polynomial by hand.

By testing these possible roots, we found that they are actually roots.

There are no other roots since the polynomial is fourth degree.
If some of the roots are irrational or complex, this process will not find them, but at least the degree of the polynomial can be reduced by dividing the factors corresponding to the rational roots.

The general solution is simply a sum of exponential functions if the roots of the characteristic equation are real.

There is a method for solving quartic equations in this book.

For more than two centuries, the question of whether similar formulas exist for the roots of higher degree equations remained open, until in 1824, when the first general solution formulas for higher degree equations were shown to be impossible.

Evariste Galois developed a more general theory in the 18th century, but it was not well known for a long time.

If this root is positive, then solutions will be infinite, if it is negative, they will be infinite.

Even though some of the roots of the characteristic equation are complex, it is still possible to express the general solution of Eq.

This term is not used in the solution, which describes an exponential decay to a steady oscillation as hows.

There are repeated roots.

If some of the roots are repeated, the solution is not the general solution of the equation.

Consider the following example.

It is possible to determine the roots of the equation by calculating the cube roots, fourth roots, or even higher roots.

The following example shows this.

The fourth roots of -1 are needed to solve the equation.

Even with computer assistance, the problem of finding all the roots of a polynomial equation may not be easy.

It may be difficult to determine if two roots are equal.
The form of the general solution in these two cases is different.

It is no longer true that the root of the characteristic equation is a complex conjugate of a root.

The solutions are complex.

The second motion has frequencies 6 and 40.

The method of undetermined coefficients is not as general as the method of variation of parameters described in the next section, but it is usually much easier to use when applicable.

The constants are determined.

The main difference in using this method is the fact that the roots of the characteristic polynomial equation may have greater than 2.

The characteristic polynomial is related to the equation.

It should be different from the other solutions of the equation.

The solution of the complete problem is the sum of the solutions of the individual component problems.

The following example shows this.

We solve the equation first.

We can come up with a solution.

If the nonhomogeneous term is even moderately complicated, the amount of algebra required to calculate the coefficients may be substantial.

A computer system can help with these calculations.

This is not possible for differential equations that do not have constant coefficients or nonhomogeneous terms.

The method of variation of parameters is discussed in the next section.

The procedure is based on the observation that sums and products of such terms can be seen as solutions of certain linear differential equations with constant coefficients.

The terms on the right side of the equation are annihilators.

By figuring out Eq.

To find the form of the solution.

The correct form of the solution can be found in the remaining terms.

Do not look at the coefficients.

To use the method of variation of parameters, it is necessary to solve the corresponding homogeneous differential equation.

Unless the coefficients are constants, this may be difficult.

To make the calculations as simple as possible, 1 conditions are chosen.

We use it.

Substituting these results in the book.