What Is a Differential Equation?

date2026-07-16document_iddoc_5eae2b535e5988ec3ae9956ee5a98a94description微分方程式を、未知の数ではなく未知関数を求める方程式として導入し、解・条件・階数・線型性・常微分と偏微分の違いを整理する。prerequisites微分法の基本 / 積分法の基本 / 導関数の定義と差商type講義content_typelecturestatusactive

mathdifferential-equationsintroductionlecture

1Introduction

This lecture explains that a differential equation has a function rather than a number as its unknown and imposes a relation between that function and its derivatives.

In the algebraic equation $x^{2} - 3 x + 2 = 0$ , the unknown $x$ is a number. By contrast, in

y^{'} = k y,

with constant $k$ , the unknown is the function $y (x)$ . The equation states a law of change: the rate of change $y^{'}$ is proportional to the value $y$ at that point. The first objective in studying differential equations is to identify both the change represented by an equation and an applicable solution method from the equation's form.

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2Terms and Definitions

An independent variable is a variable that can vary freely as an input. Ordinary differential equations commonly use $x$ or $t$ ; the latter often represents time.

An unknown function is a function to be determined so that it satisfies the equation. Examples include $y = y (x)$ and $x = x (t)$ .

The derivative $y^{'}$ describes how the unknown function changes when the independent variable changes. The occurrence of $y^{'}$ or $y^{'}'$ in a differential equation imposes a relation among a function's value, its rate of change, and the rate of change of that rate.

A solution is a function that is differentiable as many times as required on the specified interval and satisfies the equation identically when substituted into it.

For example, substituting $y = C e^{k x}$ into $y^{'} = k y$ gives

y^{'} = C k e^{k x} = k C e^{k x} = k y .

Thus, $y = C e^{k x}$ is a solution of $y^{'} = k y$ .

3What Does It Mean to Solve a Differential Equation?

To solve a differential equation is to determine its solution set on a specified interval, subject to prescribed conditions and a chosen notion of solution. In elementary standard examples, one first determines a family of functions without imposed conditions and then uses the conditions to restrict the candidates. Depending on the independence and compatibility of the conditions, there may be no solution, a unique solution, or multiple solutions.

A general solution is a family of solutions containing arbitrary constants. The general solution of $y^{'} = k y$ is

y = C e^{k x} .

A particular solution is one specific solution that satisfies the differential equation. This lecture series uses this terminology consistently.

There is more than one way to obtain a particular solution. If the general solution is known, assigning values to its arbitrary constants selects one member; for example, $C = 2$ gives $y = 2 e^{k x}$ . For a nonhomogeneous equation, one may instead construct a particular solution $y_{p}$ before forming the general solution.

A singular solution is distinct from a particular solution: it satisfies the differential equation but cannot be obtained by choosing the arbitrary constants in the general solution. A representative mechanism in nonlinear equations is the occurrence of such a solution as the envelope of a family of general solutions. Singular solutions are treated separately from the basic framework of linear operator equations.

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An initial condition specifies the value of a function or its derivatives at a reference point. For

y^{'} = k y, y (0) = y_{0},

substituting $x = 0$ into $y = C e^{k x}$ yields $C = y_{0}$ . Therefore,

y = y_{0} e^{k x}

is the solution satisfying the initial condition.

A boundary condition specifies data involving the function, its derivatives, or their linear combinations at the endpoints of an interval or on the boundary of a domain. An initial condition specifies the state at a reference point, whereas a boundary condition constrains the solution on the boundary of its domain.

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4Order of a Differential Equation

The order of a differential equation is the highest derivative of the unknown function that occurs in the equation.

The equation

y^{'} = k y

is first order because it contains no derivative higher than $y^{'}$ . The equation

m y^{'}' + c y^{'} + k y = 0

is second order because it contains $y^{'}'$ . In general, an equation containing derivatives through $y^{(n)}$ is an $n$ th-order differential equation.

Order matters because it is related to the number of conditions needed to determine a solution. For example, a second-order equation of motion often requires two initial values, position and velocity. This is a typical pattern rather than a guarantee: the independence and compatibility of the conditions must also be established.

5Linear and Nonlinear Equations

A linear differential equation is one in which the unknown function and its derivatives occur to the first power, are not multiplied by one another, and have coefficients that depend only on the independent variables.

For example,

y^{'} + p (x) y = q (x)

and

a y^{'}' + b y^{'} + c y = f (x)

are linear, whereas

y^{'} = y^{2}, y^{'}' + \sin y = 0, y y^{'} = x

are nonlinear.

Linearity matters because solutions of the associated homogeneous equation satisfy the superposition principle: linear combinations of solutions are again solutions. This principle underlies the later treatment of second-order linear and nonhomogeneous equations.

6Ordinary and Partial Differential Equations

An ordinary differential equation (ODE) has an unknown function of one independent variable. For example, if $y = y (t)$ , then

y^{'} = k y

is an ODE.

A partial differential equation (PDE) has an unknown function of two or more independent variables and contains partial derivatives. For example, if a temperature $u (t, x)$ depends on time $t$ and position $x$ , its governing equation may contain both time and spatial derivatives.

This lecture sequence focuses first on ODEs. PDEs are treated separately when spatial dependence and boundary geometry become essential.

7A Procedure for Analyzing Differential Equations

A useful initial procedure consists of problem analysis, method selection, execution, verification, and generalization.

In problem analysis, identify the unknown function, independent variables, order, and prescribed conditions.
In method selection, use the equation's form to identify candidates such as direct integration, separation of variables, or a first-order linear method.
In execution, verify the hypotheses of the selected method while carrying out the calculation.
In verification, substitute the resulting function into both the original equation and the prescribed conditions.
In generalization, determine the class of equations for which the method remains valid.

This order of analysis supports method selection from mathematical structure instead of memorization of method names.

8Minimal Example: $y^{'} = k y$

Consider $y^{'} = k y$ as an introductory example. It states that the rate of change is proportional to the current amount.

If $k > 0$ , the solutions exhibit exponential growth; if $k < 0$ , they exhibit exponential decay.

On an interval where $y \neq 0$ , rewrite the equation as

\frac{1}{y} \frac{d y}{d x} = k

and integrate to obtain

\log | y | = k x + C .

Hence,

y = C e^{k x} .

The constant solution $y = 0$ is included by taking $C = 0$ . This example illustrates the basic procedure: analyze the equation's form, select a method, and verify the resulting function by substitution.

9Subsequent Pages

data/lecture/math/differential-equations/initial-and-boundary-value-problems.lecture.n.md data/lecture/math/differential-equations/classifying-first-order-odes.lecture.n.md

10Exercise

data/exercise/math/differential-equations/introduction-to-odes-and-direct-integration.exercise.n.md