CHAPTER 3

Higher Order Differential Equations. The Laplace Transform and Special Types of Equations

Ordinary High-Order Equations

An ordinary linear differential equation of order n has the following general form:

If the function f (x) is identically zero, the equation is called homogeneous. Otherwise, the equation is called non-homogeneous. If the functions a_i(x) (i = 1, …, n) are constant, the equation is said to have constant coefficients.

A concept of great importance in this context is that of a set of linearly independent functions. A set of functions { f₁(x), f₂(x), …, f_n(x)} is linearly independent if, for any x in their common domain of definition, the Wronskian determinant of the functions is non-zero. The Wronskian determinant of the given set of functions, at a point x of their common domain of definition, is defined as follows:

The MATLAB command maple(‘Wronskian’) allows you to calculate the Wronskian matrix of a set of functions. Its syntax is:

• maple(‘Wronskian(V,x)’): This computes the Wronskian matrix corresponding to the vector of functions V with independent variable x.

A set S = { f₁(x), …, f_n(x)} of linearly independent non-trivial solutions of a homogeneous linear equation of order n

is called a set of fundamental solutions of the equation.

If the functions a_i(x) (i = 1, …, n) are continuous in an open interval I, then the homogeneous equation has a set of fundamental solutions S = {f_i(x)}  in I.

In addition, the general solution of the homogeneous equation will then be given by the function:

where {c_i} is a set of arbitrary constants.

The equation:

is called the characteristic equation of the homogeneous differential equation with constant coefficients. The solutions of this characteristic equation determine the general solutions of the corresponding differential equation.

Linear Higher-Order Equations. Homogeneous Equations with Constant Coefficients

The homogeneous linear differential equation of order n

is said to have constant coefficients if the functions  a_i(x) (i = 1, …, n) are all constant (i.e. they do not depend on the variable x).

The equation:

is called the characteristic equation of the above differential equation. The solutions (m₁, m₂, …, m_n) of this characteristic equation determine the general solution of the associated differential equation.

If the m_i (i = 1, …, n) are all different, the general solution of the homogeneous equation with constant coefficients is:

where  c₁, c₂, …, c_n are arbitrary constants.

If some m_i is a root of multiplicity k of the characteristic equation, then it determines the following k terms of the solution:

If the characteristic equation has a complex root m_j = a+bi , then its complex conjugate m_j+1 = a-bi is also a root.  These two roots determine a pair of terms in the general solution of the homogeneous equation:

MATLAB directly applies this method to obtain the solutions of homogeneous linear equations with constant coefficients, using the command dsolve or maple(‘dsolve’).

Non-Homogeneous Equations with Constant Coefficients. Variation of Parameters

Consider the non-homogeneous linear equation with constant coefficients:

Suppose { y₁(x), y₂(x),....., y_n(x)} is a linearly independent set of solutions of the corresponding homogeneous equation:

A particular solution of the non-homogeneous equation is given by:

where the functions u_i(x) are obtained as follows:

Here W_i[ y₁(x), y₂(x), …, y_n(x)] is the determinant of the matrix obtained by replacing the i-th column of the Wronskian matrix W[ y₁(x), y₂(x), …, y_n(x)] by the transpose of the vector (0, 0,…, 0, 1).

The solution of the non-homogeneous equation is then given by combining the general solution of the homogeneous equation with the particular solution of the non-homogeneous equation. If the roots m_i  of the characteristic equation of the homogeneous equation are all different, the general solution of the non-homogeneous equation is:

If some of the roots are repeated, we refer to the general form of the solution of a homogeneous equation discussed earlier.

Non-Homogeneous Equations with Variable Coefficients. Cauchy–Euler Equations

A non-homogeneous linear equation with variable coefficients of the form

is called a Cauchy–Euler equation.

MATLAB can solve this type of equation directly with the command dsolve or maple(‘dsolve’).

The Laplace Transform

Suppose f (t) is a function defined in the interval (0, ∞). The Laplace transform of f(t) is the function F(s) defined by:

We say that f (t) is the inverse Laplace transform of F(s), so that

MATLAB provides the commands maple(‘laplace’) and maple(‘invlaplace’) to calculate the Laplace transform and inverse Laplace transform of an expression with respect to a variable. Its syntax is as follows:

• maple(‘laplace(expression, t, s)’): This calculates the Laplace transform of a given expression with respect to t. The transformed variable is s.
• maple(‘(expression, s, t) invlaplace’): This computes the inverse Laplace transform of the given expression with respect to s. The inverse variable is t.

Here are some examples.

» pretty(maple('laplace(t^(3/2)-exp(t)+sinh(a*t), t, s)'));

                                1/2
                              pi        1        a
                          3/4 ----- - ----- + -------
                                5/2   s - 1    2    2
                               s              s  - a

» pretty(maple('invlaplace(s^2/(s^2+a^2)^(3/2), s, t)'))

- t BesselJ(1, a t) a + BesselJ(0, a t)

The Laplace transform and its inverse are used to solve certain differential equations. The method is to calculate the Laplace transform of each term of the equation to obtain a new differential equation, which we then solve. Finally, we find the solution of the original equation by applying the inverse Laplace transform to the solution just found.

MATLAB provides the ‘laplace’ option in the maple(‘dsolve’) command, which forces the program to solve the differential equation using the Laplace transform method. The syntax is as follows:

maple('dsolve(equation, func(var), 'laplace')')

Orthogonal Polynomials

Two functions f (x) and g(x) are said to be orthogonal on an interval [a, b] if their inner product is 0, i.e. if

An example of an orthogonal family of functions (i.e. such that any two distinct functions in the family are orthogonal) is given by:

f_n(x) = sin(nx) and g_n(x) = cos(nx), n = 1, 2, 3, …, in the interval [−π, π].

MATLAB provides a broad list of orthogonal polynomials, which are very useful in solving certain non-linear differential equations of higher order. The functions that allow us to work with these polynomials are the following:

• T(n,x), Chebychev polynomials of the first kind.
• U(n,x), Chebychev polynomials of the second kind.
• P(n,x), Legendre polynomials.
• H(n,x), Hermite polynomials.
• L(n,x), Laguerre polynomials.
• L(n,a,x), Generalized Laguerre polynomials.
• P(n,a,b,x), Jacobi polynomials.
• G(n,m,x), Gegenbauer polynomials.

Now let us look at their relationship to differential equations. It is precisely this relationship which allows us to find solutions of some non-linear equations of higher order. To use these functions we first need to run maple(‘with orthopoly’).

Chebychev Polynomials of the First and Second Kind

The Chebychev polynomials of the first kind are defined as the solutions T_n(x) of the differential equation:

Their orthogonality is given by the weighted inner product:

The Chebychev polynomials of the second kind U_n(x) are special cases of the Jacobi polynomials (see below) with a = b = 1/2. They satisfy the orthogonality relationship:

Legendre Polynomials

The Legendre polynomials P_n(x) are solutions of the Legendre differential equation:

Their orthogonality is given by the relation

Associated Legendre Polynomials

The solutions of the differential equation

are called associated Legendre polynomials.

Their orthogonality is given by the relation

where d_{k. l} is the Kronecker delta.

Hermite Polynomials

The solutions H_n(x) of the Hermite differential equation

are known as Hermite polynomials.

Their orthogonality is given by the weighted inner product:

Generalized Laguerre Polynomials

The solutions L_n(x) of the general Laguerre differential equation

are known as generalized Laguerre polynomials.

Their orthogonality is given by the weighted inner product:

Laguerre Polynomials

The solutions of the Laguerre differential equation

are known as Laguerre polynomials. This is the particular case a = 0 of the generalized Laguerre polynomials.

Jacobi Polynomials

The solutions of the Jacobi differential equation

are known as Jacobi polynomials.

Their orthogonality is given by the weighted inner product:

Gegenbauer Polynomials

The Gegenbauer polynomial G(n, a, x) is defined as follows:

The solutions of the Gegenbauer differential equation

are known as Gegenbauer polynomials.

Their orthogonality is given by the weighted inner product:

Bessel and Airy Functions

The linearly independent solutions of the following second order differential equation are called Airy functions:

(Airy equation)

The linearly independent solutions of the following second order differential equation are called Bessel functions:

(Bessel equation)

The linearly independent solutions of the following differential equation are called modified Bessel functions:

(modified Bessel equation)

MATLAB implements the following related functions:

• Ai(z) and Bi(z) are the linearly independent solutions of the Airy differential equation.
• BesselJ(n,z) and BesselY(n,z) are the linearly independent solutions of the Bessel differential equation.
• BesselI(n,z) and BesselK(n,z) are the linearly independent solutions of the modified Bessel differential equation.