### 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**

[latex]\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}[/latex]

## Glossary

- asymptotically semi-stable solution
- [latex]y=k[/latex] if it is neither asymptotically stable nor asymptotically unstable

- asymptotically stable solution
- [latex]y=k[/latex] if there exists [latex]\epsilon >0[/latex] such that for any value [latex]c\in \left(k-\epsilon ,k+\epsilon \right)[/latex] the solution to the initial-value problem [latex]{y}^{\prime }=f\left(x,y\right),y\left({x}_{0}\right)=c[/latex] approaches [latex]k[/latex] as [latex]x[/latex] approaches infinity

- asymptotically unstable solution
- [latex]y=k[/latex] if there exists [latex]\epsilon >0[/latex] such that for any value [latex]c\in \left(k-\epsilon ,k+\epsilon \right)[/latex] the solution to the initial-value problem [latex]{y}^{\prime }=f\left(x,y\right),y\left({x}_{0}\right)=c[/latex] never approaches [latex]k[/latex] as [latex]x[/latex] 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 [latex]y=c[/latex], where [latex]c[/latex] 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 [latex]h[/latex] that is added to the [latex]x[/latex] value at each step in Euler’s Method