Euler's Method (ODE)
Euler's method is a first-order numerical procedure for approximating solutions to ordinary differential equations given an initial value. Starting from a known point, the method steps forward along the tangent line to the solution curve in increments of step size h, using the ODE's derivative expression to determine the direction. Although simple to implement, Euler's method accumulates error with larger step sizes and has been superseded by more accurate methods (such as Runge–Kutta), but it remains the foundational teaching example for numerical ODE solvers.
Key Formula
y_(n+1) = y_n + h · f(x_n, y_n)
LaTeX: y_{n+1} = y_n + h \cdot f(x_n,\, y_n)
| Symbol | Meaning | Unit |
|---|---|---|
| y_n | Approximate value of y at step n | dimensionless |
| h | Step size (increment in x) | dimensionless |
| f(x_n, y_n) | Derivative dy/dx evaluated at (x_n, y_n) | dimensionless |
| y_(n+1) | Approximate value of y at next step | dimensionless |
Worked Example
Problem
Use Euler's method with h = 0.1 to approximate y(0.3) for dy/dx = x + y, y(0) = 1.
Solution
Step 0: x₀ = 0, y₀ = 1. f(0, 1) = 0 + 1 = 1. Step 1: y₁ = 1 + 0.1·1 = 1.1; x₁ = 0.1. f(0.1, 1.1) = 0.1 + 1.1 = 1.2. Step 2: y₂ = 1.1 + 0.1·1.2 = 1.22; x₂ = 0.2. f(0.2, 1.22) = 0.2 + 1.22 = 1.42. Step 3: y₃ = 1.22 + 0.1·1.42 = 1.362; x₃ = 0.3. (Exact solution y = 2e^x − x − 1 gives y(0.3) ≈ 1.3997; Euler error ≈ 0.038.)
Answer
y(0.3) ≈ 1.362 (Euler approximation)
Euler's Method Iteration Table for dy/dx = x + y, y(0) = 1, h = 0.1
| n | xₙ | yₙ | f(xₙ, yₙ) | yₙ₊₁ |
|---|---|---|---|---|
| 0 | 0.0 | 1.000 | 1.000 | 1.100 |
| 1 | 0.1 | 1.100 | 1.200 | 1.220 |
| 2 | 0.2 | 1.220 | 1.420 | 1.362 |
| 3 | 0.3 | 1.362 | 1.662 | 1.528 |
| 4 | 0.4 | 1.528 | 1.928 | 1.721 |
Interactive Tools
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Related Terms
Differential Equation
A differential equation is an equation that relates a function to one or more of its derivatives, describing how a quantity changes with respect to one or more independent variables. Ordinary differential equations (ODEs) involve functions of a single variable, while partial differential equations (PDEs) involve functions of multiple variables. Differential equations are fundamental to modeling physical, biological, and engineering systems — from Newton's laws of motion to population dynamics and heat conduction.
Linear Differential Equation
A linear differential equation is an ODE in which the unknown function and all its derivatives appear linearly (to the first power only, with no products between them), with coefficients that may be functions of the independent variable. The standard first-order linear form is dy/dx + P(x)y = Q(x), solved using an integrating factor. Linear differential equations obey the superposition principle, making them tractable analytically and foundational in engineering, electrical circuit analysis, and quantum mechanics.
Separable Differential Equation
A separable differential equation is a first-order ODE in which the variables can be algebraically rearranged so that all terms involving the dependent variable appear on one side and all terms involving the independent variable appear on the other side, allowing direct integration of each side independently. The standard form is dy/dx = g(x)·h(y), which rearranges to (1/h(y))dy = g(x)dx. Separable equations model exponential growth, radioactive decay, Newton's law of cooling, and many other phenomena.
Named after the Swiss mathematician Leonhard Euler (1707–1783), who described the method in his 1768 work "Institutiones Calculi Integralis." The method extends the tangent line approximation concept that Euler pioneered in his study of numerical analysis.