Euler’s Method

Definition

Euler's Method is a numerical technique used to approximate solutions of first-order differential equations. It uses a given initial value and steps through the domain using a fixed step size.

5 Must Know Facts For Your Next Test

1. Euler's Method approximates the solution by iterating from an initial point using the formula $y_{n+1} = y_n + h f(x_n, y_n)$.
2. The step size $h$ affects the accuracy of the approximation; smaller $h$ generally results in better accuracy.
3. Euler's Method can be applied to both linear and nonlinear differential equations.
4. Errors in Euler's Method are typically proportional to the step size, leading to global errors that are $O(h)$.
5. The method relies on direction fields to provide visual insights into how solutions will behave.

Review Questions

• What is the general formula used in Euler's Method for numerical approximation?
• How does changing the step size $h$ affect the accuracy of Euler's Method?
• Explain why direction fields are useful when applying Euler's Method.

Related terms

Differential Equation:

An equation involving derivatives that describes how a quantity changes with respect to another.

Initial Value Problem:

A differential equation along with a specified value at a given point, used as a starting point for finding solutions.

Runge-Kutta Methods: A family of more advanced numerical methods providing greater accuracy than Euler's Method for solving differential equations.