SafekipediaDifferential equation
In mathematics, a differential equation is an equation that connects one or more unknown functions and how they change, called their derivatives. These special equations help us understand how things grow or move.
Differential equations are useful in many parts of science and daily life. They are used in engineering to build bridges and machines, in physics to explain how things move, in economics to guess how markets will change, and in biology to study how groups of living things grow or shrink.
Solving a differential equation means finding a function that matches the rules the equation sets. Sometimes we can find the exact answer with math formulas. Other times, we use computers to get good guesses. Scientists and engineers use many smart ways to study these answers and see how they act over time.
History
Differential equations started when Isaac Newton and Gottfried Leibniz invented calculus. Newton talked about these equations in 1671, using special methods to solve them.
Later, other mathematicians built on this work. In 1695, Jacob Bernoulli created a special type called the Bernoulli differential equation, and Leonhard Euler helped find solutions. We also learned about important equations from studying real-world problems, like how vibrating strings in musical instruments move. This led to the wave equation. The work of Joseph-Louis Lagrange and Leonhard Euler also helped us understand mechanics with the Euler–Lagrange equation. In 1822, Fourier studied heat flow and introduced the heat equation. This equation helps us learn how heat moves through different materials.
Example
In classical mechanics, we describe how objects move by looking at their position and speed over time. Newton's laws show us how to use special math rules, called differential equations, to connect an object's position to time.
One example is figuring out how fast a ball falls when we think about gravity and air pushing back on it. The ball's speed changes because of gravity and air resistance, and we can describe this using a differential equation. Solving this equation helps us learn how the ball's speed changes as it falls.
Types
Differential equations can be grouped in many ways. This helps us find the best way to solve them.
One common way is to see if the equation is ordinary or partial. An ordinary differential equation involves functions of just one variable and their derivatives. A partial differential equation involves functions of several variables and their partial derivatives.
Another way to classify them is as linear or non-linear. Linear differential equations have the unknown function and its derivatives appearing to the first power and not multiplied together. Non-linear equations do not follow this rule. Many important equations in physics, like those describing radioactive decay or heat transfer, are linear.
Initial conditions and boundary conditions
When solving a differential equation, we often need extra information to find the exact solution. This extra information is called a condition.
If we are looking at how something moves over time, these conditions are called initial conditions. They tell us where and how fast an object starts.
If we are looking at how something behaves in space, like a vibrating string, these conditions are called boundary conditions. They tell us what happens at the ends of the string.
The number of these conditions must match the complexity of the equation we are solving.
Existence of solutions
For a differential equation, we often want to know if a solution exists and if it is only one solution.
One important result is the Peano existence theorem. It tells us that for certain conditions, there will be a solution to a simpler type of differential equation called a first-order initial value problem.
For more complicated equations, called linear initial value problems, we need more conditions. If the functions involved are continuous over an interval, then a solution exists and is also unique.
Related concepts
Some related ideas in math include delay differential equations, where the rate of change depends on past values. Integral equations are similar but use sums over time instead of rates of change. There are also stochastic differential equations, which involve chance processes, and stochastic partial differential equations that add space to these chance processes. Other types include integro-differential equations that mix rates of change with sums, and differential algebraic equations that combine both rates of change and direct relationships.
Connection to difference equations
Differential equations are related to difference equations. In difference equations, the variable only has certain values. The equation links the value of an unknown function at one point to its values at nearby points. Many ways to solve differential equations, like the Euler method, use difference equations to estimate the solution.
Applications
Differential equations are important in many areas of science and engineering. They are used in pure and applied mathematics, physics, and engineering. These equations help us understand and predict natural processes. Examples include the movement of planets, the design of bridges, and how neurons work in our brains. Often, these equations do not have simple answers. So, scientists use special methods called numerical methods to find approximate solutions.
Many basic laws in physics and chemistry can be written as differential equations. In fields like biology and economics, they help us study complicated systems and how they change over time. Interestingly, different problems from various sciences can lead to the same differential equations. For example, the way light, sound, and water waves move can all be described by the same equation called the wave equation. Similarly, the way heat spreads can be understood using another important equation known as the heat equation. This shows how differential equations can bring together many different natural phenomena under one mathematical idea.
Main article: List of named differential equations
Software
Some special computer programs can help solve differential equations. These programs use commands to find answers.
Here are the commands used in popular programs:
This article is a child-friendly adaptation of the Wikipedia article on Differential equation, available under CC BY-SA 4.0.