Essential Concepts
- A direction field is a mathematical object used to graphically represent solutions to a first-order differential equation.
- Euler’s Method is a numerical technique that can be used to approximate solutions to a differential equation.
Key Equations
- Euler’s Method
Glossary
- asymptotically semi-stable solution
- if it is neither asymptotically stable nor asymptotically unstable
- asymptotically stable solution
- if there exists such that for any value the solution to the initial-value problem approaches as approaches infinity
- asymptotically unstable solution
- if there exists such that for any value the solution to the initial-value problem never approaches as approaches infinity
- direction field (slope field)
- a mathematical object used to graphically represent solutions to a first-order differential equation; at each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point
- equilibrium solution
- any solution to the differential equation of the form , where is a constant
- Euler’s Method
- a numerical technique used to approximate solutions to an initial-value problem
- solution curve
- a curve graphed in a direction field that corresponds to the solution to the initial-value problem passing through a given point in the direction field
- step size
- the increment that is added to the value at each step in Euler’s Method
Candela Citations
CC licensed content, Shared previously
- Calculus Volume 2. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-2/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction