Homogeneous and Non-Homogeneous
An ordinary differential equation (ODE) constitutes a mathematical relationship between a function of a single independent variable, its derivatives, and the variable itself.
Such equations are used in formulating physical laws and modeling temporal evolution in numerous scientific disciplines.
They describe systems where the rate of change of a state is determined by the state itself. This exposition provides a formal definition and a classification of ordinary differential equations.
Definition: let I \subseteq \mathbb{R} be an open interval. We consider a function y: I \to \mathbb{R} that is at least n-times differentiable. An ordinary differential equation imposes a constraint on the values of the independent variable x, the function y(x), and its first n derivatives, denoted as y'(x), y''(x), \ldots, y^{(n)}(x).
The most general representation of an n^{th} order ODE is the implicit form. It is given by a function F that maps a subset of \mathbb{R}^{n+2} to \mathbb{R}, such that the equation is expressed as:
F\left(x, y(x), \frac{\mathrm d y}{\mathrm d x}, \frac{\mathrm d^2 y}{\mathrm d x^2}, \ldots, \frac{\mathrm d^n y}{\mathrm d x^n}\right) = 0
This formulation states that for any x \in I, the tuple consisting of x and the function y evaluated at x along with its derivatives up to order n must belong to the zero set of the function F.
In many applications, it is possible to isolate the highest-order derivative. This leads to the explicit form of an n^{th} order ODE:
\frac{\mathrm d^n y}{\mathrm d x^n} = f\left(x, y(x), \frac{\mathrm d y}{\mathrm d x}, \ldots, \frac{\mathrm d^{n-1} y}{\mathrm d x^{n-1}}\right)
Here, the function f is defined on a domain in \mathbb{R}^{n+1}.
The order of the differential equation is determined by n, the highest derivative present in the expression.
The structural properties of the function F or f allow for a classification of ODEs, which in turn informs the methods applicable for their solution.
A differential equation is classified as linear if the function F is a linear function of y and its derivatives.
An n^{th} order linear ODE can be written in the canonical form:
a_n(x) \frac{\mathrm d^n y}{\mathrm d x^n} + a_{n-1}(x) \frac{\mathrm d^{n-1} y}{\mathrm d x^{n-1}} + \cdots + a_1(x) \frac{\mathrm d y}{\mathrm d x} + a_0(x) y(x) = r(x)
It is conventional to assume a_n(x) \neq 0 on the interval of interest, allowing one to write the equation in normalized explicit form:
\frac{\mathrm d^n y}{\mathrm d x^n} = \sum_{i=0}^{n-1} p_i(x) \frac{\mathrm d^i y}{\mathrm d x^i} + q(x)
The functions p_i(x) and q(x) are not required to be linear in x. The linearity of the equation is solely with respect to the dependent variable y and its derivatives. This structure permits the application of the principle of superposition.
We can define a linear differential operator L as:
L[y] = \frac{\mathrm d^n y}{\mathrm d x^n} - \sum_{i=0}^{n-1} p_i(x) \frac{\mathrm d^i y}{\mathrm d x^i}
The ODE can then be compactly written as L[y] = q(x).
Any ODE that cannot be expressed in the linear form described above is classified as nonlinear.
The principle of superposition does not hold for nonlinear equations, which makes their analysis more complex.
Nonlinear equations can exhibit a wide range of behaviors not seen in linear systems, such as chaos, finite-time escape, and limit cycles.
The nature of the term q(x), often called the source term, leads to a further sub-classification.
If q(x) = 0 for all x in the domain, the equation is homogeneous, If q(x) \neq 0, the equation is termed non-homogeneous.
An homogeneous linear ODE takes the form:
L[y] = 0
The set of solutions to a homogeneous linear ODE forms a vector space of dimension n. A basis for this solution space can be constructed, and any linear combination of these basis solutions is also a solution.
The general solution, often denoted y_c(x), is such a linear combination with n arbitrary constants. The trivial solution, y(x) = 0, is always a solution to a homogeneous equation.
A non-homogeneous linear ODE takes the form:
L[y] = q(x)
The general solution to a nonhomogeneous equation is found by summing a particular solution, y_p(x), with the general solution of the corresponding homogeneous equation, y_c(x).
y(x) = y_c(x) + y_p(x)
The set of all solutions to the nonhomogeneous equation forms an affine space, which is a translation of the vector space of homogeneous solutions by a particular solution vector.
A differential equation is designated as autonomous if the independent variable x does not explicitly appear in the function F or f. For an explicit n^{th} order equation, this means:
\frac{\mathrm d^n y}{\mathrm d x^n} = f\left(y(x), \frac{\mathrm d y}{\mathrm d x}, \ldots, \frac{\mathrm d^{n-1} y}{\mathrm d x^{n-1}}\right)
Autonomous equations are prevalent in physics and engineering where the independent variable is time, and the physical laws governing the system do not change over time.
Their analysis often involves the study of the phase space of the system.
Often, physical or mathematical models involve multiple interdependent quantities. This leads to a system of coupled differential equations.
Let \mathbf{y}(x) be a vector-valued function with m components:
\mathbf{y}(x) = [y_1(x), y_2(x), \dots, y_m(x)]^T
An explicit system of m coupled n^{th} order ODEs takes the form:
\frac{\mathrm d^n \mathbf{y}}{\mathrm d x^n} = \mathbf{F}\left(x, \mathbf{y}, \frac{\mathrm d \mathbf{y}}{\mathrm d x}, \ldots, \frac{\mathrm d^{n-1} \mathbf{y}}{\mathrm d x^{n-1}}\right)
where \mathbf{F} is a vector-valued function with m components.
Any n^{th} order ODE can be converted into a system of n first-order ODEs.
Let’s consider the equation:
y^{(n)} = f(x, y, y^\prime, \ldots, y^{(n-1)})
We can define a vector of state variables \mathbf{z}(x) as:
\mathbf{z}(x) = \begin{bmatrix} z_1(x) \\ z_2(x) \\ \vdots \\ z_n(x) \end{bmatrix} = \begin{bmatrix} y(x) \\ y^\prime(x) \\ \vdots \\ y^{(n-1)}(x) \end{bmatrix}
The dynamics of this state vector are governed by the first-order system:
\frac{\mathrm d \mathbf{z}}{\mathrm d x} = \begin{bmatrix} z_2 \\ z_3 \\ \vdots \\ f(x, z_1, z_2, \ldots, z_n) \end{bmatrix}
This reduction to a first-order system is useful for the theoretical analysis of existence and uniqueness, as well as for the development of numerical algorithms.
A function u: I \to \mathbb{R}, where I is an open interval, is a solution to an n^{th} order ODE if u is n-times continuously differentiable on I and satisfies the equation for all x \in I.
The general solution of an n^{th} order ODE is a family of solutions characterized by n independent arbitrary constants. To specify a unique solution, one must impose additional constraints. These typically take the form of initial conditions or boundary conditions.
A particular solution is a single solution obtained from the general solution by assigning specific values to these constants. For an Initial Value Problem (IVP), all conditions are specified at a single point, x_0 \in I:
\begin{bmatrix} y(x_0) = y_0, \\ y^\prime(x_0) = y_1 \\ \ldots \\ y^{(n-1)}(x_0) = y_{n-1} \end{bmatrix}
Picard-Lindelöf theorem (here) provides conditions (namely, Lipschitz continuity of f) for the local existence and uniqueness of a solution to such an IVP.
A singular solution is a solution that cannot be obtained from the general solution by any choice of the arbitrary constants. Such solutions often arise as envelopes to the family of curves representing the general solution.
A solution u: I \to \mathbb{R} is a maximal solution if it cannot be extended to a solution on any larger interval J \supset I.
If the domain of a solution is the entire real line, I = \mathbb{R}, it is called a global solution.
The study of whether solutions are global or if they go to infinity in finite time is an important topic in the qualitative theory of differential equations.