---
title: "Partial Differential Equation | Linear Algebra"
description: "A partial differential equation is an equation for a multivariable function and its partial derivatives, used in Linear Algebra and Differential Equations to model heat, waves, and flow."
canonical: "https://fiveable.me/linear-algebra-and-differential-equations/key-terms/partial-differential-equation"
type: "key-term"
subject: "Linear Algebra and Differential Equations"
unit: "Unit 7"
---

# Partial Differential Equation | Linear Algebra

## Definition

A partial differential equation, or PDE, is an equation involving an unknown function of several variables and its partial derivatives. In Linear Algebra and Differential Equations, PDEs model change across space and time.

## What It Is

A partial differential equation is an equation that connects a function of several variables with one or more of its partial derivatives. In this course, that usually means you are tracking how something changes in more than one direction at once, such as temperature across a metal rod over time or the shape of a vibrating string across space and time.

The word partial matters because the derivative is taken with respect to one variable while the others are treated like constants. That is different from an ordinary differential equation, which has derivatives with respect to just one independent variable. A PDE can describe a whole field, surface, or region instead of a single changing quantity.

A classic example is the heat equation, where temperature depends on both position and time. The equation tells you how heat flows from warmer regions to cooler ones. A wave equation does something similar for motion, like a plucked string or sound wave, while Laplace's equation often shows up when the system is in a steady state.

In Linear Algebra and Differential Equations, PDEs are usually not solved by a single algebra trick. You often separate variables, turn the PDE into ordinary differential equations, and then use boundary conditions or initial conditions to pin down the specific solution. That is where the linear algebra side starts to matter, because eigenvalues, eigenfunctions, and linear combinations often organize the solution.

One common mistake is to treat a PDE like a regular formula you can isolate immediately. Most PDEs describe families of possible solutions, and the extra conditions decide which one fits the physical situation. If a problem gives you a region, a starting state, or fixed edges, those details are part of the PDE setup, not extra decoration.

## Why It Matters

PDEs are the language for problems where change depends on more than one variable at a time. In this course, that makes them a bridge between differential equations and linear algebra, because the methods you use often rely on structured solution spaces, matrix ideas, and special functions.

They show up anytime a system spreads, bends, vibrates, or diffuses. Temperature moving through a material, fluid flow, and waves on a string all need equations that describe behavior across space and time together, not one variable at a time.

PDEs also force you to think about conditions on the edges of a problem. A solution is not just something that satisfies the equation, it also has to fit the initial or boundary data. That is why the same PDE can produce very different answers in different settings.

This term also prepares you for the way later topics are organized. Classification into elliptic, parabolic, and hyperbolic equations tells you what kind of behavior to expect and which methods are likely to work. So when you see a PDE, you are not just seeing a complicated equation, you are seeing a map of how a system evolves or stays in balance.

## Connections

### Ordinary Differential Equation

An ordinary differential equation uses derivatives with respect to one independent variable, while a partial differential equation uses two or more. That difference changes the whole solution process. In this course, ODE methods often show up inside PDE techniques, especially after separation of variables turns a big multivariable problem into smaller single-variable equations.

### [Boundary Conditions](/linear-algebra-and-differential-equations/key-terms/boundary-conditions)

Boundary conditions tell a PDE what happens at the edges of the region you are studying. Without them, you usually get too many possible solutions. For heat, waves, or steady-state problems, boundary conditions can represent fixed temperatures, insulated edges, or anchored endpoints.

### Initial Value Problem

An initial value problem gives the starting state of a system, usually at time zero. For a PDE, the initial condition works with the equation to narrow the solution down to the behavior that matches the real situation. This is common in time-dependent models like heat flow or vibration.

### [Fourier Series](/linear-algebra-and-differential-equations/key-terms/fourier-series)

Fourier series often appear when solving PDEs with separation of variables. They let you build a solution from simpler sine and cosine pieces that match boundary conditions. In practice, this is one of the main ways a complicated PDE solution gets written in a usable form.

## On the AP Exam

A quiz or problem set question on a partial differential equation usually asks you to identify the independent variables, classify the equation, or match it to the right model. You might be asked whether a formula is a PDE or an ODE, or to say what kind of physical process it represents. In a worked problem, the next move is often to check the boundary or initial conditions, then choose a method like separation of variables.

If the equation comes from a word problem, your job is to translate the situation into a function of space and time, not just write symbols. For example, a heat problem may ask where the temperature is changing fastest, or a wave problem may ask what the endpoints are doing. The grading usually focuses on whether you set up the variables and conditions correctly, not just whether you recognize the term.

## Partial Differential Equation vs Ordinary Differential Equation

These are easy to mix up because both involve derivatives, but they describe different kinds of change. An ordinary differential equation has one independent variable, like time. A partial differential equation has at least two, so you take partial derivatives and usually need boundary or initial data to get a specific solution.

## Key Takeaways

- A partial differential equation is an equation for a function of several variables, and it uses partial derivatives to describe how that function changes.
- In this course, PDEs usually model systems that depend on space and time together, like heat, waves, or steady-state behavior.
- You often need boundary conditions or initial conditions to pick out the right solution from many possibilities.
- Separation of variables, Fourier series, and related methods often turn a PDE into simpler ordinary differential equations.
- The type of PDE, such as elliptic, parabolic, or hyperbolic, gives you clues about what the system does and how to solve it.

## FAQs

### What is a partial differential equation in Linear Algebra and Differential Equations?

A partial differential equation is an equation that involves an unknown multivariable function and its partial derivatives. In this course, it usually describes how something changes across space and time, like temperature, vibration, or fluid flow. The main idea is that more than one independent variable affects the outcome.

### How is a partial differential equation different from an ordinary differential equation?

An ordinary differential equation has derivatives with respect to one independent variable, while a PDE has derivatives with respect to two or more. That means PDEs usually need more setup, such as boundary conditions or initial conditions. Many PDE methods also reduce the problem to a system of ODEs.

### What are examples of partial differential equations?

Common examples include the heat equation, the wave equation, and Laplace's equation. The heat equation models diffusion of temperature, the wave equation models vibration or sound, and Laplace's equation often describes steady-state situations. These show up in physics and engineering more than in pure algebra-style problems.

### How do you solve a partial differential equation in class?

A common first step is separation of variables, especially when the problem has clear boundary conditions. From there, you may get ordinary differential equations that can be solved and combined into a series solution, often with Fourier series. Some PDEs are also solved numerically when a closed-form answer is messy or unavailable.

## Related Study Guides

- [7.1 Basic Concepts and Classifications of Differential Equations](/linear-algebra-and-differential-equations/unit-7/basic-concepts-classifications-differential-equations/study-guide/d8qoivESWsbiUnAH)
- [7.3 Modeling with Differential Equations](/linear-algebra-and-differential-equations/unit-7/modeling-differential-equations/study-guide/ovcp0vE5dMw19CaT)

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