chapter 9 math104e definitions

A ____ involes an unknown function y and its derivative.

Differential equation

The ___ of a differential equation is the highest order that appears on a derivative in the equation

order

A differential equation is ____ if the unknown function y and its derivative appear only to the first power and are not composed with other functions

linear

The general ____ has the form ____ where p and q are given functions of t

first order linear differential equation, dy/dt + p(t)y = g(t)

The most general family of functions that solves a differential equation, including the appropriate number of arbitrary constants, is called the ____

general solution

A differential equation is often accompanied by ____ that specify the values of y, and possibly its derivatives, at a particular point

initial condition

A differential equation together with the appropriate number of initial conditions, is called an_____

initial value problem

A _____ to the first order initial value problem y’(t) = F(t,y), y(a) = b is a function y(t) that satisfies the differential equation on an interval I (the domain of the solution) and whose graph includes the point (a,b)

solution

A _____ is a picture that shows the slope of the solution at selected points in the ty-plane

direction field

A _____ is a first-order differential equation in which the expression y’(t) = F(t,y) can be written in the form g(y)y’(t) = h(t)

separable equation

The differential equation y’(t) = f(y) has a (constant) ____ solution y = a when f(a) = 0

equilibrium

The equilibrium is ____ if the initial conditions near y = a produce solutions that approach y = a as t → infinity

stable

The equilibrium is ____ if the initial conditions near y = a produce solutions that do not approach y = a as t → infinity

unstable

The ___ f specifies the rate of growth of the population and is chosen to give the best description of the population

growth rate function

For a logistic model, the population increases, but not without bound, instead it approaches equilibrium, or ____, solutions of P = 0 and P = K

steady-state

For a logistic model, the equilibrium population is called the ____ P = K

carrying capacity