Calculus Volume 2

# Key Terms

### Key Terms

asymptotically semi-stable solution
$y=ky=k$ if it is neither asymptotically stable nor asymptotically unstable
asymptotically stable solution
$y=ky=k$ if there exists $ε>0ε>0$ such that for any value $c∈(k−ε,k+ε)c∈(k−ε,k+ε)$ the solution to the initial-value problem $y′=f(x,y),y(x0)=cy′=f(x,y),y(x0)=c$ approaches $kk$ as $xx$ approaches infinity
asymptotically unstable solution
$y=ky=k$ if there exists $ε>0ε>0$ such that for any value $c∈(k−ε,k+ε)c∈(k−ε,k+ε)$ the solution to the initial-value problem $y′=f(x,y),y(x0)=cy′=f(x,y),y(x0)=c$ never approaches $kk$ as $xx$ approaches infinity
autonomous differential equation
an equation in which the right-hand side is a function of $yy$ alone
carrying capacity
the maximum population of an organism that the environment can sustain indefinitely
differential equation
an equation involving a function $y=y(x)y=y(x)$ 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 $y=c,y=c,$ where $cc$ 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 $r>0r>0$ in the exponential growth function $P(t)=P0ertP(t)=P0ert$
initial population
the population at time $t=0t=0$
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 $t=0t=0$
initial-value problem
a differential equation together with an initial value or values
integrating factor
any function $f(x)f(x)$ 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 $a(x)y′+b(x)y=c(x)a(x)y′+b(x)y=c(x)$
logistic differential equation
a differential equation that incorporates the carrying capacity $KK$ and growth rate $rr$ 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 $y′=f(x)g(y)y′=f(x)g(y)$
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 $y=f(x)y=f(x)$ 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 $y′+p(x)y=q(x)y′+p(x)y=q(x)$
step size
the increment $hh$ that is added to the $xx$ value at each step in Euler’s Method
threshold population
the minimum population that is necessary for a species to survive
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