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- Mathematics LibreTexts - Introduction to Differential Equations (Oct. 17, 2026)
differential equation
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What is a differential equation?
A differential equation is a mathematical equation showing the derivatives of functions. It is a statement that contains one or more derivatives representing the rates of change of continuously varying quantities.
Differential equations are common in science and engineering because rates of change can be directly observed and measured for systems undergoing changes. The solution to a differential equation is an equation expressing the functional dependence of one variable on one or more others. This solution produces a function that predicts the behavior of the original system within certain constraints.
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How is an ordinary differential equation different from other equations?
An ordinary differential equation (ODE) is a mathematical equation that relates a function of one variable to its derivatives. The key difference between ODEs and other types of differential equations lies in the number of independent variables involved. ODEs involve functions depending on only a single variable, whereas partial differential equations involve functions that depend on several independent variables. The order of a differential equation corresponds to the highest-order derivative it contains. Solutions to differential equations provide a function for predicting the behavior of a system.
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What is a partial differential equation?
A partial differential equation relates a function of multiple variables to its partial derivatives. A partial derivative shows how a function changes as one variable changes while others remain constant.
Many important partial differential equations in physics are second-order and linear. Examples include the Laplace equation, the heat equation, and the wave equation.
Partial differential equations are generally more challenging to solve. However, techniques have been developed for solving simpler classes of equations called linear, and for classes known loosely as "almost" linear equations.
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How are differential equations used to model real-world problems?
Differential equations model real-world problems by describing the rates of change of quantities. They are common in science and engineering because rates of change can be directly observed and measured in systems undergoing changes.
For example, exponential decay can be modeled using differential equations, with applications ranging from radioactive decay to Newton's law of cooling. In these cases, differential equations relate the rates of change of quantities to their current values, making it possible to predict future behavior. The solutions must satisfy a differential equation and auxiliary conditions, such as boundary values, that describe the influence from outside a region.
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What does it mean to solve a differential equation?
Solving a differential equation involves deducing a solution from the equation using algebra and calculus, or integrating the equation. The solution is typically an equation that expresses the functional dependence of one variable on others and includes constant terms not found in the original differential equation. In essence, solving a differential equation produces a function that predicts the behavior of the original system within certain constraints.
However, most differential equations can't be explicitly solved, requiring indirect study methods. When a direct solution is impossible, numerical analysis methods using computers can be employed to find approximate solutions. A solution of a differential equation may satisfy auxiliary conditions such as boundary values or initial values.
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What is an initial value problem in differential equations and why is it important?
In differential equations, an initial value problem involves finding a solution that satisfies both the differential equation and a set of initial conditions. These conditions specify the value of the function and its derivatives at a particular point, representing the system's compressed history and influencing its future behavior.
Initial value problems are important because they provide a unique solution to the differential equation. The solutions of differential equations generally involve unspecified constants that are determined by auxiliary conditions. This allows for predicting the behavior of a system under specific conditions, which is essential in many scientific and engineering applications. Numerical methods are often employed to find approximate solutions.
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differential equation, mathematical statement containing one or more derivativesāthat is, terms representing the rates of change of continuously varying quantities. Differential equations are very common in science and engineering, as well as in many other fields of quantitative study, because what can be directly observed and measured for systems undergoing changes are their rates of change. The solution of a differential equation is, in general, an equation expressing the functional dependence of one variable upon one or more others; it ordinarily contains constant terms that are not present in the original differential equation. Another way of saying this is that the solution of a differential equation produces a function that can be used to predict the behaviour of the original system, at least within certain constraints.
Differential equations are classified into several broad categories, and these are in turn further divided into many subcategories. The most important categories are ordinary differential equations and partial differential equations. When the function involved in the equation depends on only a single variable, its derivatives are ordinary derivatives and the differential equation is classed as an ordinary differential equation. On the other hand, if the function depends on several independent variables, so that its derivatives are partial derivatives, the differential equation is classed as a partial differential equation. The following are examples of ordinary differential equations: 
In these, y stands for the function, and either t or x is the independent variable. The symbols k and m are used here to stand for specific constants.
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Whichever the type may be, a differential equation is said to be of the nth order if it involves a derivative of the nth order but no derivative of an order higher than this. The equation 
is an example of a partial differential equation of the second order. The theories of ordinary and partial differential equations are markedly different, and for this reason the two categories are treated separately.
Instead of a single differential equation, the object of study may be a simultaneous system of such equations. The formulation of the laws of dynamics frequently leads to such systems. In many cases, a single differential equation of the nth order is advantageously replaceable by a system of n simultaneous equations, each of which is of the first order, so that techniques from linear algebra can be applied.
An ordinary differential equation in which, for example, the function and the independent variable are denoted by y and x is in effect an implicit summary of the essential characteristics of y as a function of x. These characteristics would presumably be more accessible to analysis if an explicit formula for y could be produced. Such a formula, or at least an equation in x and y (involving no derivatives) that is deducible from the differential equation, is called a solution of the differential equation. The process of deducing a solution from the equation by the applications of algebra and calculus is called solving or integrating the equation. It should be noted, however, that the differential equations that can be explicitly solved form but a small minority. Thus, most functions must be studied by indirect methods. Even its existence must be proved when there is no possibility of producing it for inspection. In practice, methods from numerical analysis, involving computers, are employed to obtain useful approximate solutions.