AP Calculus: Introduction to Differential Equations flashcards

Question 1

What is an equation called that relates an unknown function y of x to one or more of its derivatives?

Accepted Answer

A differential equation

Question 2

What is a function y = f(x) called when it satisfies a given differential equation?

Accepted Answer

A solution to the differential equation

Question 3

How is the order of a differential equation determined?

Accepted Answer

It is the highest order of derivative of the unknown function that shows up in the equation.

Question 4

What name is given to the complete set of solutions of a differential equation?

Accepted Answer

The general solution (a family of solutions)

Question 5

What is a differential equation paired with one or more initial values called?

Accepted Answer

An initial-value problem

Question 6

A differential equation whose right-hand side depends on y alone is called what?

Accepted Answer

An autonomous differential equation

Question 7

What is a solution of the form y = c, with c a constant, called?

Accepted Answer

An equilibrium solution

Question 8

A solution y = k is called what if some ε greater than 0 exists such that every starting value c within ε of k gives an initial-value solution that tends to k as x grows without bound?

Accepted Answer

Asymptotically stable

Question 9

What graphical tool for a first-order differential equation draws, at each point, a short segment with the slope of the solution through that point?

Accepted Answer

A direction field (slope field)

Question 10

In a direction field, what is the curve that follows the solution of the initial-value problem through a chosen point called?

Accepted Answer

A solution curve

Question 11

What numerical technique approximates solutions of an initial-value problem?

Accepted Answer

Euler's Method

Question 12

In Euler's Method, what is the increment h added to x at every step called?

Accepted Answer

The step size

Question 13

What is a function called that you multiply into both sides of a differential equation so the side containing the unknown function becomes the derivative of a product?

Accepted Answer

An integrating factor

Question 14

A first-order differential equation that can be put in the form a(x)y′ + b(x)y = c(x) is called what?

Accepted Answer

A linear differential equation

Question 15

What is the form y′ + p(x)y = q(x) of a first-order linear differential equation called?

Accepted Answer

Standard form

Question 16

An equation that can be written as y′ = f(x)g(y) is called what?

Accepted Answer

A separable differential equation

Question 17

What method solves a separable differential equation?

Accepted Answer

Separation of variables

Question 18

What is the name of the differential equation that brings the carrying capacity K and the growth rate r into a population model?

Accepted Answer

The logistic differential equation

Question 19

What is the largest population of a species that its environment can support indefinitely called?

Accepted Answer

The carrying capacity

Question 20

In the exponential growth function P(t) = P₀e^(rt), what is the positive constant r called?

Accepted Answer

The growth rate

Question 21

What is the population at time t = 0 called?

Accepted Answer

The initial population

Question 22

What visual summary shows how solutions of an autonomous differential equation behave for various initial conditions?

Accepted Answer

A phase line

AP Calculus: Introduction to Differential Equations flashcards

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