### Key Terms

- 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 ${y}^{\prime}=f\left(x,y\right),\phantom{\rule{1em}{0ex}}y\left({x}_{0}\right)=c$ 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 ${y}^{\prime}=f\left(x,y\right),\phantom{\rule{1em}{0ex}}y\left({x}_{0}\right)=c$ never approaches $k$ as $x$ approaches infinity

- autonomous differential equation
- an equation in which the right-hand side is a function of $y$ alone

- carrying capacity
- the maximum population of an organism that the environment can sustain indefinitely

- differential equation
- an equation involving a function $y=y(x)$ and one or more of its derivatives

- 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

- general solution (or family of solutions)
- the entire set of solutions to a given differential equation

- growth rate
- the constant $r>0$ in the exponential growth function $P\left(t\right)={P}_{0}{e}^{rt}$

- initial population
- the population at time $t=0$

- initial value(s)
- a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable

- initial velocity
- the velocity at time $t=0$

- initial-value problem
- a differential equation together with an initial value or values

- integrating factor
- any function $f\left(x\right)$ that is multiplied on both sides of a differential equation to make the side involving the unknown function equal to the derivative of a product of two functions

- linear
- description of a first-order differential equation that can be written in the form $a\left(x\right){y}^{\prime}+b\left(x\right)y=c\left(x\right)$

- logistic differential equation
- a differential equation that incorporates the carrying capacity $K$ and growth rate $r$ into a population model

- order of a differential equation
- the highest order of any derivative of the unknown function that appears in the equation

- particular solution
- member of a family of solutions to a differential equation that satisfies a particular initial condition

- phase line
- a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions

- separable differential equation
- any equation that can be written in the form $y\prime =f\left(x\right)g\left(y\right)$

- separation of variables
- a method used to solve a separable differential equation

- 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

- solution to a differential equation
- a function $y=f(x)$ that satisfies a given differential equation

- standard form
- the form of a first-order linear differential equation obtained by writing the differential equation in the form $y\prime +p\left(x\right)y=q\left(x\right)$

- step size
- the increment $h$ that is added to the $x$ value at each step in Euler’s Method

- threshold population
- the minimum population that is necessary for a species to survive