Linear
differential equationsA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
are of the form
-
where the
differential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
L is a linear operator, y is the unknown function (such as a function of time y(t)), and the right hand side ƒ is a given function of the same nature as y (called the
source term). For a function dependent on time we may write the equation more expressively as
-
and, even more precisely by bracketing
-
The linear operator L may be considered to be of the form
-
The linearity condition on L rules out operations such as taking the square of the derivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of y; but permits, for example, taking the second derivative of y.
It is convenient to rewrite this equation in an operator form
-
where D is the differential operator d/dt (i.e. Dy = y' , D2y = y",... ), and the An are given functions.
Such an equation is said to have order n, the index of the highest derivative of y that is involved.
A typical simple example is the linear differential equation used to model radioactive decay. Let N(t) denote the number of radioactive atoms in some sample of material at time t. Then for some constant k > 0, the number of radioactive atoms which decay can be modelled by
If y is assumed to be a function of only one variable, one speaks about an ordinary differential equationIn mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
, else the derivatives and their coefficients must be understood as (contractedIn multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor caused by applying the summation...
) vectors, matrices or tensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s of higher rank, and we have a (linear) partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
.
The case where ƒ = 0 is called a homogeneous equation and its solutions are called complementary functions. It is particularly important to the solution of the general case, since any complementary function can be added to a solution of the inhomogeneous equation to give another solution (by a method traditionally called particular integral and complementary function). When the Ai are numbers, the equation is said to have constant coefficientsIn mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. In other words, it singles out special operators, within the larger class of...
.
Homogeneous equations with constant coefficients
The first method of solving linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form 
, for possibly-complex values of 
. The exponential function is one of the few functions that keep its shape after differentiation. In order for the sum of multiple derivatives of a function to sum up to zero, the derivatives must cancel each other out and the only way for them to do so is for the derivatives to have the same form as the initial function. Thus, to solve
we set 
, leading to
Division by e zx gives the nth-order polynomial
This algebraic equation F(z) = 0, is the characteristic equationCharacteristic equation may refer to:* Characteristic equation , used to solve linear differential equations* Characteristic equation, a characteristic polynomial equation in linear algebra used to find eigenvalues...
considered later by Gaspard MongeGaspard Monge, Comte de Péluse was a French mathematician, revolutionary, and was inventor of descriptive geometry. During the French Revolution, he was involved in the complete reorganization of the educational system, founding the École Polytechnique...
and Augustin-Louis Cauchy.
Formally, the terms
of the original differential equation are replaced by zk. Solving the polynomial gives n values of z, z1, ..., zn. Substitution of any of those values for z into e zx gives a solution e zix. Since homogeneous linear differential equations obey the superposition principleIn physics and systems theory, the superposition principle , also known as superposition property, states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually...
, any linear combinationIn mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of these functions also satisfies the differential equation.
When these roots are all distinct, we have n distinct solutions to the differential equation. It can be shown that these are linearly independent, by applying the Vandermonde determinant, and together they form a basisIn linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
of the space of all solutions of the differential equation.
The preceding gave a solution for the case when all zeros are distinct, that is, each has multiplicity 1. For the general case, if z is a (possibly complex) zero (or root) of F(z) having multiplicity m, then, for
,
is a solution of the ODE. Applying this to all roots gives a collection of n distinct and linearly independent functions, where n is the degree of F(z). As before, these functions make up a basis of the solution space.
If the coefficients A
i of the differential equation are real, then real-valued solutions are generally preferable. Since non-real roots z then come in
conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
pairs, so do their corresponding basis functions , and the desired result is obtained by replacing each pair with their real-valued
linear combinationIn mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
s Re(y) and Im(y), where y is one of the pair.
A case that involves complex roots can be solved with the aid of
Euler's formulaEuler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
.
Examples
Given
. The characteristic equation is
which has roots 2+i and 2−i. Thus the solution basis
is
. Now y is a solution if and only if
for
.
Because the coefficients are real,
- we are likely not interested in the complex solutions
- our basis elements are mutual conjugates
The linear combinations
and
will give us a real basis in
.
Simple harmonic oscillator
The second order differential equation
which represents a simple
harmonic oscillatorIn classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....
, can be restated as
The expression in parenthesis can be factored out, yielding
which has a pair of linearly independent solutions, one for
and another for
The solutions are, respectively,
and
These solutions provide a basis for the two-dimensional "
solution spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
" of the second order differential equation: meaning that linear combinations of these solutions will also be solutions. In particular, the following solutions can be constructed
and
These last two trigonometric solutions are linearly independent, so they can serve as another basis for the solution space, yielding the following general solution:
Damped harmonic oscillator
Given the equation for the damped
harmonic oscillatorIn classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....
:
the expression in parentheses can be factored out: first obtain the characteristic equation by replacing D with λ. This equation must be satisfied for all y, thus:
Solve using the quadratic formula:
Use these data to factor out the original differential equation:
This implies a pair of solutions, one corresponding to
and another to
The solutions are, respectively,
and
where ω = b / 2m. From this linearly independent pair of solutions can be constructed another linearly independent pair which thus serve as a basis for the two-dimensional solution space:
However, if |ω| < |ω
0| then it is preferable to get rid of the consequential imaginaries, expressing the general solution as
This latter solution corresponds to the underdamped case, whereas the former one corresponds to the overdamped case: the solutions for the underdamped case
oscillateOscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
whereas the solutions for the overdamped case do not.
Nonhomogeneous equation with constant coefficients
To obtain the solution to the
nonhomogeneous equation (sometimes called
inhomogeneous equation), find a particular integral y
P(x) by either the
method of undetermined coefficientsIn mathematics, the method of undetermined coefficients, also known as the lucky guess method, is an approach to finding a particular solution to certain inhomogeneous ordinary differential equations and recurrence relations...
or the
method of variation of parametersIn mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations...
; the general solution to the linear differential equation is the sum of the general solution of the related homogeneous equation and the particular integral. Or, when the initial conditions are set, use
Laplace transform to obtain the particular solution directly.
Suppose we face
For later convenience, define the characteristic polynomial
We find the solution basis
as in the homogeneous (f(x)=0) case. We now seek a
particular integral y
p(x) by the
variation of parameters method. Let the coefficients of the linear combination be functions of x:
For ease of notation we will drop the dependency on x (i.e. the various (x)). Using the "operator" notation
and a broad-minded use of notation, the ODE in question is
; so
With the constraints
the parameters commute out, with a little "dirt":
But
, therefore
This, with the constraints, gives a linear system in the
. This much can always be solved; in fact, combining
Cramer's ruleIn linear algebra, Cramer's rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution...
with the
WronskianIn mathematics, the Wronskian is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes be used to show that a set of solutions is linearly independent.-Definition:...
,
The rest is a matter of integrating
The particular integral is not unique;
also satisfies the ODE for any set of constants c
j.
Example
Suppose
. We take the solution basis found above
.
Using the
list of integrals of exponential functions
And so
(Notice that u
1 and u
2 had factors that canceled y
1 and y
2; that is typical.)
For interest's sake, this ODE has a physical interpretation as a driven damped
harmonic oscillatorIn classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....
; y
p represents the steady state, and
is the transient.
Equation with variable coefficients
A linear ODE of order n with variable coefficients has the general form
Examples
A simple example is the Cauchy–Euler equation often used in engineering
First order equation
A linear ODE of order 1 with variable coefficients has the general form
Where D is the
differential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
. Equations of this form can be solved by multiplying the
integrating factorIn mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus, in this case often multiplying through by an...
throughout to obtain
which simplifies due to the
product ruleIn calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
to
-
which, on integrating both sides, yields
-
-
In other words: The solution of a first-order linear ODE
-
with coefficients that may or may not vary with x, is:
where
is the constant of integration, and
-
A compact form of the general solution is (see J. Math. Chem. 48 (2010) 175):
-
where
is the generalized Dirac delta function.
Examples
Consider a first order differential equation with
constant coefficientsIn mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. In other words, it singles out special operators, within the larger class of...
:
This equation is particularly relevant to first order systems such as
RC circuitA resistor–capacitor circuit ', or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source...
s and
mass-damperIn physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...
systems.
In this case, p(x) = b, r(x) = 1.
Hence its solution is
See also
- Continuous-repayment mortgage
- Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
- Laplace transform
- List of differentiation identities, Nth Derivatives Section
External links