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 $y^{\prime }=f\left(x,y\right),y\left(x_0\right)=c$ 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

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