Table of Contents
Overview
Differential equations are equations that involve an unknown function and one or more of its derivatives. They arise naturally whenever we model quantities that change: population growth, cooling of an object, motion of a planet, electric circuits, and much more.
In this chapter, we build a gentle, conceptual introduction to differential equations as a subject, without going deeply into any particular solving technique (those are handled in the later subsections such as “First-Order Differential Equations” and “Second-Order Differential Equations”). Here, the focus is on:
- What a differential equation is
- The different basic types
- What it means to be a solution
- Why initial and boundary conditions matter
- How differential equations are used for modeling change
What Is a Differential Equation?
A differential equation is an equation that relates a function to its derivatives. The unknown is not just a number but an entire function.
For example, suppose $y$ is a function of $x$. Then each of the following is a differential equation:
- $$\frac{dy}{dx} = 3x^2$$
- $$\frac{dy}{dx} = y$$
- $$\frac{d^2y}{dx^2} + y = 0$$
- $$x \frac{dy}{dx} + 2y = \sin x$$
In each case:
- $y = y(x)$ is the unknown function.
- Derivatives like $\frac{dy}{dx}$ or $\frac{d^2y}{dx^2}$ appear.
- The equation links $y$, its derivatives, and possibly the variable $x$.
A central question in differential equations is:
Given a differential equation, can we find all functions $y(x)$ that satisfy it?
Those functions are called the solutions of the differential equation.
Order and Degree
Two basic descriptors of a differential equation are its order and degree.
Order
The order of a differential equation is the highest derivative that appears.
Examples:
- $$\frac{dy}{dx} = 3x^2$$
Only the first derivative $\frac{dy}{dx}$ appears, so this is a first-order differential equation. - $$\frac{d^2y}{dx^2} + y = 0$$
The highest derivative is $\frac{d^2y}{dx^2}$, so this is a second-order differential equation. - $$\frac{d^3y}{dx^3} - 4\frac{dy}{dx} + y = 0$$
The highest derivative is third order, so it is a third-order differential equation.
Degree (informal idea)
When the equation is polynomial in the derivatives (no roots, no denominators, no trig of derivatives, etc.), the degree is the highest power of the highest-order derivative.
For instance:
- $$\left(\frac{dy}{dx}\right)^2 + y = 0$$
First order (highest derivative is $\frac{dy}{dx}$) and degree $ (the derivative is squared). - $$\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 = 0$$
Second order; degree $ (the highest order derivative appears to the first power, but there is a first derivative cubed).
In an introductory course, the order is much more important than the degree, and we will mostly focus on first-order and second-order differential equations in later sections.
Ordinary vs Partial Differential Equations
There are two main types of differential equations, depending on what kind of derivatives appear.
Ordinary Differential Equations (ODEs)
An ordinary differential equation uses derivatives with respect to a single independent variable (often $x$ or $t$).
Typical examples:
- $$\frac{dy}{dx} = y$$
- $$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 3y = 0$$
Here $y$ is a function of one variable (say $x$), written $y(x)$, and derivatives are ordinary derivatives.
Most of this course’s “Differential Equations” material focuses on ODEs.
Partial Differential Equations (PDEs)
A partial differential equation involves a function of several variables and its partial derivatives.
For example, if $u = u(x,t)$ depends on both position $x$ and time $t$, then
$$
\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}
$$
is a partial differential equation (often called the heat equation). The symbols $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial t}$ denote partial derivatives.
PDEs are essential for modeling waves, heat, fluids, and many physical systems in multiple dimensions, but they are more advanced. In this course, the focus is on ODEs; PDEs are mainly mentioned for context.
Solutions of Differential Equations
A solution to a differential equation is a function that makes the equation true when it is substituted in, together with its derivatives.
Verifying a Solution
To check if a function is a solution:
- Compute the necessary derivatives.
- Substitute the function and its derivatives into the differential equation.
- See if the equation holds for all values in the domain of interest.
Example: Consider the equation
$$
\frac{dy}{dx} = y.
$$
Try the function $y(x) = e^x$:
- $\dfrac{dy}{dx} = e^x$.
- Substituting into the equation:
$$
\frac{dy}{dx} = e^x,\quad y = e^x.
$$
Both sides match, so $y = e^x$ is a solution.
Notice that $y = Ce^x$ (where $C$ is any constant) also works:
- $\dfrac{dy}{dx} = Ce^x$,
- $y = Ce^x$,
- again $\frac{dy}{dx} = y$.
So the differential equation $\frac{dy}{dx} = y$ has infinitely many solutions, one for each constant $C$.
This is typical: many differential equations have entire families of solutions.
General vs Particular Solutions
Because solving a differential equation often introduces constants, we distinguish:
- A general solution: a family of solutions containing one or more arbitrary constants.
- A particular solution: a specific solution obtained from the general solution by choosing specific values of the constants, often determined by extra conditions (like initial values).
Example:
- General solution of $\frac{dy}{dx} = y$:
$$
y(x) = Ce^x,
$$
where $C$ is any real constant. - If we know that $y(0) = 5$ (extra information), we find $C$:
$$
y(0) = Ce^0 = C = 5,
$$
so the particular solution is $y(x) = 5e^x$.
Initial and Boundary Conditions
A differential equation alone usually does not determine a unique solution. To model a specific physical situation, we must add extra information, such as the value of the function at a given point.
These extra conditions are called initial conditions or boundary conditions, depending on the context.
Initial Value Problems (IVPs)
When the independent variable represents time (often denoted $t$), it is natural to specify the state of the system at some starting time $t_0$.
An initial value problem consists of:
- A differential equation, and
- Values of the function (and sometimes its derivatives) at a particular initial point.
Example:
- Differential equation:
$$
\frac{dy}{dt} = ky,
$$
where $k$ is a constant. - Initial condition:
$$
y(0) = y_0.
$$
Together they form the initial value problem
$$
\frac{dy}{dt} = ky,\quad y(0) = y_0.
$$
Solving the differential equation gives the general solution $y(t) = Ce^{kt}$. Using the initial condition:
$$
y_0 = y(0) = Ce^{k\cdot 0} = C,
$$
so $C = y_0$, and the unique solution that matches the initial condition is
$$
y(t) = y_0 e^{kt}.
$$
Boundary Value Problems (BVPs)
Sometimes the independent variable represents a spatial coordinate (like position along a rod), and conditions are given at more than one point. These are boundary conditions, and the problem is called a boundary value problem.
A simple example is a second-order equation describing the shape of a hanging chain, or the temperature along a rod with fixed temperatures at both ends.
The important idea here is:
- Initial value: data specified at one point (often in time).
- Boundary value: data specified at one or more endpoints of an interval (often in space).
Both types of problems are crucial in applications.
Linear vs Nonlinear Differential Equations
Another important distinction is between linear and nonlinear differential equations.
Linear Differential Equations (Conceptual)
An ordinary differential equation is called linear in the unknown function $y$ if:
- $y$ and its derivatives appear only to the first power,
- $y$ and its derivatives are not multiplied together,
- there are no nonlinear functions of $y$ or its derivatives (like $\sin y$, $y^2$, $e^y$, etc.).
A first-order linear ODE in $y$ typically looks like
$$
a_1(x)\frac{dy}{dx} + a_0(x)y = g(x),
$$
where $a_1(x)$, $a_0(x)$, and $g(x)$ are given functions of $x$.
Example:
- $$\frac{dy}{dx} + 2y = \sin x$$ is linear.
A second-order linear ODE usually looks like
$$
a_2(x)\frac{d^2y}{dx^2} + a_1(x)\frac{dy}{dx} + a_0(x)y = g(x).
$$
Example:
- $$\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y = 0$$ is linear.
Linear equations have special properties and systematic solution methods, which will appear in the later subsections.
Nonlinear Differential Equations
If an equation is not linear, it is nonlinear. That happens if $y$ (or its derivatives) shows up with powers other than $1$, products of derivatives appear, or nonlinear functions of $y$ are present.
Examples:
- $$\left(\frac{dy}{dx}\right)^2 + y = 0$$
- $$\frac{dy}{dx} = y^2$$
- $$\frac{d^2y}{dx^2} + y^3 = 0$$
Nonlinear differential equations can be much harder to solve and often require numerical methods or qualitative analysis.
Direction Fields and Qualitative Behavior (First-Order)
For first-order equations, even when an exact solution is difficult to find, we can often get a picture of how solutions behave using a direction field (also called a slope field).
Consider a first-order ODE written as
$$
\frac{dy}{dx} = f(x,y).
$$
At each point $(x,y)$ in the plane, the equation tells us the slope of a solution curve passing through that point: the slope is $f(x,y)$. A direction field is built by drawing small line segments with that slope at many points. The solution curves then “follow” the pattern of these slopes.
You do not need to construct direction fields in detail here; the important idea is:
- A differential equation determines the tangent slope of its solutions at each point.
- From that, we can sketch how solutions behave without solving them exactly.
This kind of qualitative view is a powerful tool, especially for nonlinear equations.
Differential Equations as Models of Change
One major reason differential equations are important is that many real-world processes can be described by stating how fast a quantity changes, rather than giving its value directly.
In words:
A differential equation relates the rate of change of a quantity to the quantity itself (and possibly other variables).
A few common types of situations:
- Population growth: The rate of change of a population is often assumed to be proportional to its current size, leading to equations like
$$
\frac{dP}{dt} = kP.
$$ - Cooling/heating: The rate of temperature change of an object can be proportional to the difference between its temperature and the surrounding temperature.
- Motion under forces: Newton’s second law, $F = ma$, leads to second-order differential equations, because acceleration $a$ is the second derivative of position with respect to time.
- Radioactive decay: The rate of decay of a substance is proportional to how much of it remains, giving an equation similar to population growth but with $k < 0$.
In each case, the key steps are:
- Express a physical principle (such as “rate of change is proportional to amount”) as a differential equation.
- Solve the equation (if possible) or analyze its behavior.
- Interpret the solution in terms of the real-world situation.
Details of these applications and specific solution methods are developed in the subsections “First-Order Differential Equations,” “Second-Order Differential Equations,” and “Applications.”
Existence and Uniqueness (Informal Idea)
When we solve a differential equation with given initial conditions, it is natural to ask:
- Does a solution exist?
- If it exists, is it unique?
For many well-behaved equations, the answer is “yes” under reasonable conditions. There are theorems (often called existence and uniqueness theorems) that give precise conditions guaranteeing:
- At least one solution exists near the initial point.
- There is at most one solution that satisfies the given initial conditions.
At an intuitive level, this means that for many “reasonable” differential equations, if you specify:
- The equation itself, and
- Enough initial or boundary data,
then the behavior of the system is determined completely (no ambiguity).
You do not need the formal theorems at this stage, but it is useful to know that this question is an important part of the theory.
What Comes Next in This Course
This chapter has introduced the broad ideas:
- What differential equations are,
- How they are classified (by order, linearity, and type: ODE vs PDE),
- What solutions mean and why initial/boundary conditions matter,
- Why differential equations are central to modeling change.
In the following chapters in this section:
- First-Order Differential Equations: You will learn specific methods for solving several key types of first-order ODEs (such as separation of variables).
- Second-Order Differential Equations: You will see how to handle second-order linear equations, which often model vibrations, circuits, and more.
- Applications: You will explore concrete physical and population models and how differential equations are used to analyze them.