Last modified: April 26, 2022

This article is written in: 🇺🇸

Ordinary Differential Equations (ODEs)

An ordinary differential equation (ODE) is an equation that involves:

I. One independent variable, often denoted by t (in many applications, t represents time).

II. One dependent variable (or unknown function), which we may denote by y(t).

III. The derivatives of the dependent variable with respect to the independent variable.

Formally, an ODE can be written as

F(t,y(t),y′(t),y″

where y^{(n)}(t) denotes the n-th derivative of y with respect to t. The integer n is called the order of the ODE.

The term ordinary differentiates it from a partial differential equation (PDE), in which derivatives with respect to multiple independent variables can appear.

Classification and Terminology

• Order: The order of an ODE is the order of the highest derivative appearing in the equation.
• Degree: The degree of an ODE is the exponent of the highest-order derivative after the equation has been simplified and cleared of any fractional or irrational expressions in the derivatives.
• Linearity vs. Nonlinearity:

A (single) ODE is linear if it can be expressed in the form

a_n(t)y^{(n)}(t) + a_{n-1}(t)y^{(n-1)}(t) + \cdots + a_1(t)y'(t) + a_0(t)y(t) = g(t)

where a_0(t), \dots, a_n(t) and g(t) are functions of t only (i.e., do not depend on y or its derivatives). If any product of the dependent variable and/or its derivatives or any power other than 1 of y or its derivatives appears, then the ODE is nonlinear.

• Homogeneous vs. Nonhomogeneous (Inhomogeneous) for Linear ODEs: A linear ODE is called homogeneous if g(t) \equiv 0. Otherwise, it is nonhomogeneous or inhomogeneous if g(t) \neq 0.
• Autonomous vs. Nonautonomous: An autonomous ODE is one in which the independent variable t does not appear explicitly in the function F. For instance, y' = f(y) is autonomous. A nonautonomous ODE explicitly depends on t, for example y' = f(t, y).

Understanding Differential Equations

Differential equations capture the relationship between a function (representing a quantity of interest) and its rates of change. These arise naturally in numerous domains:

• In physics, Newton’s laws of motion lead to second-order ODEs in time describing the positions of objects.
• In biology, population growth can often be modeled by first-order ODEs.
• In engineering, circuits and systems obey Kirchhoff’s laws or mass-spring-damper systems described by second-order ODEs.
• In economics, growth models and dynamic systems can be formulated as ODEs.

Initial Conditions and Boundary Conditions

To find a unique solution, one often needs:

• Initial conditions, e.g., for a first-order ODE y'(t) = f(t,y), one typically specifies y(t_0) = y_0.
• For higher-order ODEs, more initial values (or boundary values) are required. For example, a second-order ODE might need y(t_0) = y_0 and y'(t_0) = v_0.

The combination of a differential equation and enough conditions to fix a unique solution is called an initial value problem (IVP) or boundary value problem (BVP), depending on whether the conditions are specified at a single point (IVP) or at different points (BVP).

Existence and Uniqueness of Solutions

A crucial theoretical aspect of ODEs is ensuring whether a solution to a given IVP exists and whether it is unique. One fundamental result for first-order ODEs is the Picard–Lindelöf theorem (also known as the Existence and Uniqueness Theorem), which states that if:

I. f(t,y) is continuous in a region around (t_0, y_0),

II. f satisfies a Lipschitz condition in y (i.e., there exists a constant L such that

\bigl| f(t, y_1) - f(t, y_2) \bigr| \le L \bigl| y_1 - y_2 \bigr|

for all y_1, y_2 in that region), then there exists a time interval (t_0 - \delta, t_0 + \delta) on which there is a unique solution y(t) satisfying y(t_0) = y_0.

Main Concepts in Ordinary Differential Equations

We revisit the key concepts with further mathematical detail:

Order

If an ODE contains derivatives up to the n-th derivative, it is called an n-th order ODE. For example:

• \frac{dy}{dt} = f(t, y) is a first-order ODE.
• \frac{d^2y}{dt^2} + a(t)\frac{dy}{dt} + b(t)y = 0 is a second-order ODE.

Degree

The degree is determined by writing the ODE in polynomial form in its highest-order derivative. For example, the ODE

\left(y''\right)^2 + \left(y'\right)^3 + y = 0 is not in polynomial form due to the squared second derivative and cubed first derivative. However, if we could (somehow) algebraically solve for the highest-order derivative and rewrite it linearly or as a polynomial expression without fractional exponents, then the exponent of that highest-order derivative would tell us the degree. Many ODEs (especially linear ones) are understood in simpler terms: the degree is typically 1 for linear ODEs.

Linearity

A linear ODE of order n has the form

a_n(t)y^{(n)} + a_{n-1}(t)y^{(n-1)} + \cdots + a_1(t)y' + a_0(t)y = g(t),

where each a_k(t) (for k = 0,1,\dots,n) and g(t) depend only on t. No products like (y')^2 or yy'' occur, nor do terms such as \sin(y). If such terms do occur, the ODE is nonlinear.

Homogeneity

• A linear ODE is homogeneous if g(t) = 0. The homogeneous form is

a_n(t)y^{(n)} + a_{n-1}(t)y^{(n-1)} + \cdots + a_1(t)y' + a_0(t)y = 0. - It is nonhomogeneous (or inhomogeneous) if g(t) \neq 0.

Autonomous ODEs

• An autonomous ODE does not explicitly depend on t. Formally, it takes a form such as y' = f(y).
• The solutions and their qualitative behavior can often be studied using phase-line analysis (for first-order autonomous ODEs) or phase-plane analysis (for second-order systems), etc.

Mathematical Forms of Ordinary Differential Equations

Below are some standard forms that frequently appear.

General First-Order ODE

\frac{dy}{dt} = f\bigl(t, y(t)\bigr), \quad y(t_0) = y_0

• Goal: Find a function y(t) that satisfies the differential equation for t in some interval containing t_0 and also satisfies the initial condition y(t_0) = y_0.

First-Order Linear ODE

\frac{dy}{dt} + p(t)y = g(t).

This is a subset of the above form but is special because its solution technique is well-known. A classic approach is the Integrating Factor method:

I. Multiply both sides by the integrating factor

\mu(t) = e^{\int p(t)dt}.

II. Rewrite the left-hand side as the derivative of \mu(t)y(t).

III. Integrate both sides w.r.t. t to solve for y(t).

Second-Order Linear ODE

\frac{d^2y}{dt^2} + a(t)\frac{dy}{dt} + b(t)y = g(t)

• Homogeneous if g(t) = 0.
• Nonhomogeneous if g(t) \neq 0.

Constant-Coefficient Case

When a and b are constants, the ODE

y'' + ay' + by = g(t)

can be solved using:

I. Characteristic equation for the associated homogeneous part:

r^2 + ar + b = 0

II. The solution of the homogeneous ODE depends on the discriminant \Delta = a^2 - 4b:

• If \Delta > 0, two distinct real roots r_1 and r_2.
• If \Delta = 0, a repeated real root r.
• If \Delta < 0, two complex conjugate roots \alpha \pm i\beta.

III. A particular solution y_p(t) must be found (e.g., via the method of undetermined coefficients or variation of parameters) for the nonhomogeneous case.

IV. The general solution is y(t) = y_h(t) + y_p(t).

Autonomous ODE

\frac{dy}{dt} = f\bigl(y(t)\bigr).

Analyzing equilibrium (steady-state) solutions where f(y)=0 is a powerful tool for studying the long-term behavior (qualitative analysis).

Solutions of Ordinary Differential Equations

General Remarks

A solution to an ODE on an interval I is a function y(t) that:

I. Is differentiable up to the required order on I.

II. Substitutes into the ODE to satisfy it identically for all t \in I.

General vs. Particular Solutions

• A general solution often contains constants (like C_1, C_2, \ldots) that can be set by initial or boundary conditions.
• A particular solution is a single, specific solution that satisfies both the ODE and a given set of boundary/initial conditions.

Classic Examples

I. First-Order Linear with Constant Coefficient

\frac{dy}{dt} = ay \quad \longrightarrow \quad \frac{dy}{y} = adt

Integrating both sides:

\ln|y| = at + C \quad \longrightarrow \quad y(t) = C_1 e^{a t}

If y(t_0) = y_0, then C_1 = y_0 e^{-a t_0}.

II. Second-Order Homogeneous with Constant Coefficients

y'' + ay' + by = 0. The characteristic equation is r^2 + ar + b = 0. Let \Delta = a^2 - 4b. - If \Delta > 0 with roots r_1, r_2, the general solution is

y(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}. - If \Delta = 0 with repeated root r, the general solution is

y(t) = \bigl(C_1 + C_2t\bigr) e^{r t}. - If \Delta < 0 with complex roots \alpha \pm i\beta, the general solution is

y(t) = e^{\alpha t}\bigl(C_1 \cos(\beta t) + C_2 \sin(\beta t)\bigr).

III. Second-Order Nonhomogeneous with Constant Coefficients

y'' + ay' + by = g(t).

One finds the general solution as

y(t) = y_h(t) + y_p(t), where y_h(t) is the general solution of the homogeneous equation, and y_p(t) is any particular solution of the original nonhomogeneous equation.

Examples of Ordinary Differential Equations

I. Newton’s Second Law of Motion

Often written as F = ma. Since a = \frac{d^2 x}{dt^2},

m\frac{d^2x}{dt^2} = F(x, t).

This is a second-order ODE. If F depends only on x and t (and possibly v = x'), it may be nonlinear or linear (if F is linear in x, x', etc.).

II. Population Dynamics

The logistic model:

\frac{dP}{dt} = rP \left(1 - \frac{P}{K}\right), is a first-order nonlinear autonomous ODE describing population growth with a carrying capacity K.

III. RC Circuit (First-Order Linear ODE)

Consider a resistor R in series with a capacitor C. The voltage V_C(t) across the capacitor satisfies:

C\frac{dV_C}{dt} + \frac{V_C(t)}{R} = \frac{V_{\text{in}}(t)}{R}.

Rearranged:

\frac{dV_C}{dt} + \frac{1}{RC}V_C(t) = \frac{V_{\text{in}}(t)}{RC}.

This is a first-order linear ODE.

Applications

ODEs are ubiquitous in mathematical modeling across disciplines:

• Quantum mechanics uses the Schrödinger equation, which can reduce to ordinary differential equations in specific cases.
• The motion of particles and rigid bodies is described by Newton's laws or through Lagrangian/Hamiltonian formulations, involving ordinary differential equations.
• System dynamics and control theory rely on transfer functions and state-space models based on ordinary differential equations.
• Electrical circuits, such as RLC circuits and operational amplifiers, are modeled using ordinary differential equations to describe voltage and current over time.
• The spread of infectious diseases is modeled with the SIR framework, a system of ordinary differential equations for population groups.
• Biochemical reactions are often described by the Michaelis–Menten equations, which are ordinary differential equations for reaction rates.
• Macroeconomic systems use dynamical models to study growth and other phenomena through ordinary differential equations.
• Option pricing models, while typically based on partial differential equations, can simplify to ordinary differential equations under certain assumptions.
• Chemical kinetics describes reaction rates using ordinary differential equations for product formation.

Limitations and Complexities

• Analytical solutions often do not exist for many ordinary differential equations, especially nonlinear ones, requiring the use of numerical methods like Euler’s method and Runge–Kutta methods.
• High-order and nonlinear ODEs become increasingly complex to solve, leading to reliance on qualitative methods such as stability analysis and phase-plane diagrams instead of closed-form solutions.
• Simplifying assumptions underpin many ODE models, such as linearization or ignoring certain effects, but these assumptions can limit the solution's applicability when they do not hold.
• Parameter sensitivity in real-world models means small changes in parameters can result in drastically different outcomes, as seen in chaotic systems.