Linear differential equation


In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form
where and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable.
Such an equation is an ordinary differential equation. A linear differential equation may also be a linear partial differential equation, if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives.

Types of solution

A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant coefficients. An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two, Kovacic's algorithm allows deciding whether there are solutions in terms of integrals, and computing them if any.
The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. Their representation by the defining differential equation and initial conditions allows making algorithmic most operations of calculus, such as computation of antiderivatives, limits, asymptotic expansion, and numerical evaluation to any precision, with a certified error bound.

Basic terminology

The highest order of derivation that appears in a differential equation is the order of the equation. The term, which does not depend on the unknown function and its derivatives, is sometimes called the constant term of the equation, even when this term is a non-constant function. If the constant term is the zero function, then the differential equation is said to be homogeneous, as it is a homogeneous polynomial in the unknown function and its derivatives. The equation obtained by replacing, in a linear differential equation, the constant term by the zero function is the '. A differential equation has constant coefficients if only constant functions appear as coefficients in the associated homogeneous equation.
A ' of a differential equation is a function that satisfies the equation.
The solutions of a homogeneous linear differential equation form a vector space. In the ordinary case, this vector space has a finite dimension, equal to the order of the equation. All solutions of a linear differential equation are found by adding to a particular solution any solution of the associated homogeneous equation.

Linear differential operator

A basic differential operator of order is a mapping that maps any differentiable function to its th derivative, or, in the case of several variables, to one of its partial derivatives of order. It is commonly denoted
in the case of univariate functions, and
in the case of functions of variables. The basic differential operators include the derivative of order 0, which is the identity mapping.
A linear differential operator is a linear combination of basic differential operators, with differentiable functions as coefficients. In the univariate case, a linear operator has thus the form
where are differentiable functions, and the nonnegative integer is the order of the operator.
Let be a linear differential operator. The application of to a function is usually denoted or, if one needs to specify the variable. A linear differential operator is a linear operator, since it maps sums to sums and the product by a scalar to the product by the same scalar.
As the sum of two linear operators is a linear operator, as well as the product of a linear operator by a differentiable function, the linear differential operators form a vector space over the real numbers or the complex numbers. They form also a free module over the ring of differentiable functions.
The language of operators allows a compact writing for differentiable equations: if
is a linear differential operator, then the equation
may be rewritten
There may be several variants to this notation; in particular the variable of differentiation may appear explicitly or not in and the right-hand and of the equation, such as or.
The kernel of a linear differential operator is its kernel as a linear mapping, that is the vector space of the solutions of the differential equation.
In the case of an ordinary differential operator of order, Carathéodory's existence theorem implies that, under very mild conditions, the kernel of is a vector space of dimension, and that the solutions of the equation have the form
where are arbitrary numbers. Typically, the hypotheses of Carathéodory's theorem are satisfied in an interval, if the functions are continuous in, and there is a positive real number such that for every in.

Homogeneous equation with constant coefficients

A homogeneous linear differential equation has constant coefficients if it has the form
where are numbers. In other words, it has constant coefficients if it is defined by a linear operator with constant coefficients.
The study of these differential equations with constant coefficients dates back to Leonhard Euler, who introduced the exponential function, which is the unique solution of the equation, such that. It follows that the th derivative of is, and this allows solving homogeneous linear differential equations rather easily.
Let
be a homogeneous linear differential equation with constant coefficients.
Searching solutions of this equation that have the form is equivalent to searching the constants such that
Factoring out , shows that must be a root of the characteristic polynomial
of the differential equation, which is the left-hand side of the characteristic equation
When these roots are all distinct, one has distinct solutions that are not necessarily real, even if the coefficients of the equation are real. These solutions can be shown to be linearly independent, by considering the Vandermonde determinant of the values of these solutions at. Together they form a basis of the vector space of solutions of the differential equation.
Example

has the characteristic equation
This has zeros,,, and . The solution basis is thus
A real basis of solution is thus

In the case where the characteristic polynomial has only simple roots, the preceding provides a complete basis of the solutions vector space. In the case of multiple roots, more linearly independent solutions are needed for having a basis. These have the form
where is a nonnegative integer, is a root of the characteristic polynomial of multiplicity, and. For proving that these functions are solutions, one may remark that if is a root of the characteristic polynomial of multiplicity, the characteristic polynomial may be factored as. Thus, applying the differential operator of the equation is equivalent with applying first times the operator and then the operator that has as characteristic polynomial. By the exponential shift theorem,
and thus one gets zero after application of
As, by the fundamental theorem of algebra, the sum of the multiplicities of the roots of a polynomial equals the degree of the polynomial, the number of above solutions equals the order of the differential equation, and these solutions form a basis of the vector space of the solutions.
In the common case where the coefficients of the equation are real, it is generally more convenient to have a basis of the solutions consisting of real-valued functions. Such a basis may be obtained from the preceding basis by remarking that, if is a root of the characteristic polynomial, then is also a root, of the same multiplicity. Thus a real basis is obtained by using Euler's formula, and replacing and by and.

Second-order case

A homogeneous linear differential equation of the second order may be written
and its characteristic polynomial is
If and are real, there are three cases for the solutions, depending on the discriminant. In all three cases, the general solution depends on two arbitrary constants and.
  • If, the characteristic polynomial has two distinct real roots, and. In this case, the general solution is
  • If, the characteristic polynomial has a double root, and the general solution is
  • If, the characteristic polynomial has two complex conjugate roots, and the general solution is which may be rewritten in real terms, using Euler's formula as
Finding the solution satisfying and, one equates the values of the above general solution at and its derivative there to and, respectively. This results in a linear system of two linear equations in the two unknowns and. Solving this system gives the solution for a so-called Cauchy problem, in which the values at for the solution of the DEQ and its derivative are specified.

Non-homogeneous equation with constant coefficients

A non-homogeneous equation of order with constant coefficients may be written
where are real or complex numbers, is a given function of, and is the unknown function.
There are several methods for solving such an equation. The best method depends on the nature of the function that makes the equation non-homogeneous. If is a linear combination of exponential and sinusoidal functions, then the exponential response formula may be used. If, more generally, is a linear combination of functions of the form,, and, where is a nonnegative integer, and a constant, then the method of undetermined coefficients may be used. Still more general, the annihilator method applies when satisfies a homogeneous linear differential equation, typically, a holonomic function.
The most general method is the variation of constants, which is presented here.
The general solution of the associated homogeneous equation
is
where is a basis of the vector space of the solutions and are arbitrary constants. The method of variation of constants takes its name from the following idea. Instead of considering as constants, they can be considered as unknown functions that have to be determined for making a solution of the non-homogeneous equation. For this purpose, one adds the constraints
which imply
for, and
Replacing in the original equation and its derivatives by these expressions, and using the fact that are solutions of the original homogeneous equation, one gets
This equation and the above ones with as left-hand side form a system of linear equations in whose coefficients are known functions. This system can be solved by any method of linear algebra. The computation of antiderivatives gives, and then.
As antiderivatives are defined up to the addition of a constant, one finds again that the general solution of the non-homogeneous equation is the sum of an arbitrary solution and the general solution of the associated homogeneous equation.