## Summary of Direction Fields and Numerical Methods

### 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

$\begin{array}{c}{x}_{n}={x}_{0}+nh\hfill \\ {y}_{n}={y}_{n - 1}+hf\left({x}_{n - 1},{y}_{n - 1}\right),\text{where}h\text{is the step size}\hfill \end{array}$

## Glossary

asymptotically semi-stable solution
$y=k$ if it is neither asymptotically stable nor asymptotically unstable
asymptotically stable solution
$y=k$ if there exists $\epsilon >0$ such that for any value $c\in \left(k-\epsilon ,k+\epsilon \right)$ the solution to the initial-value problem ${y}^{\prime }=f\left(x,y\right),y\left({x}_{0}\right)=c$ approaches $k$ as $x$ approaches infinity
asymptotically unstable solution
$y=k$ if there exists $\epsilon >0$ such that for any value $c\in \left(k-\epsilon ,k+\epsilon \right)$ the solution to the initial-value problem ${y}^{\prime }=f\left(x,y\right),y\left({x}_{0}\right)=c$ never approaches $k$ as $x$ 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 $y=c$, where $c$ 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 $h$ that is added to the $x$ value at each step in Euler’s Method