Calculus: Introduction to Differential Equations flashcards

Question 1

What is a differential equation?

Accepted Answer

An equation linking a function y = y(x) with at least one of its derivatives

Question 2

What is the order of a differential equation?

Accepted Answer

The order of the highest derivative of the unknown function that occurs in it

Question 3

What counts as a solution of a differential equation?

Accepted Answer

Any function y = f(x) that satisfies the equation

Question 4

What is a particular solution?

Accepted Answer

A member of the family of solutions that satisfies a given initial condition

Question 5

What is an initial-value problem?

Accepted Answer

A differential equation paired with one or more initial values

Question 6

What is an equilibrium solution?

Accepted Answer

A solution of the form y = c, where c is a constant

Question 7

What is an autonomous differential equation?

Accepted Answer

One whose right-hand side is a function of y alone

Question 8

What is a direction field also called?

Accepted Answer

A slope field

Question 9

In a direction field, what slope does the line segment at each point have?

Accepted Answer

The slope of the solution to the differential equation that passes through that point

Question 10

What is a phase line?

Accepted Answer

A picture of how solutions of an autonomous differential equation behave under various initial conditions

Question 11

What is Euler's Method?

Accepted Answer

A numerical technique for approximating solutions to an initial-value problem

Question 12

In Euler's Method, what is the step size?

Accepted Answer

The increment h added to the x value at each step

Question 13

Euler's Method: y_n = ?

Accepted Answer

y_n = y_(n−1) + h·f(x_(n−1), y_(n−1)), with x_n = x₀ + nh

Question 14

What form can every separable differential equation be written in?

Accepted Answer

y′ = f(x)g(y)

Question 15

What method is used to solve a separable differential equation?

Accepted Answer

Separation of variables

Question 16

What is carrying capacity?

Accepted Answer

The largest population of an organism its environment can support indefinitely

Question 17

Logistic differential equation with initial condition?

Accepted Answer

dP/dt = rP(1 − P/K), P(0) = P₀

Question 18

Solution of the logistic initial-value problem: P(t) = ?

Accepted Answer

P(t) = P₀K e^(rt) / ((K − P₀) + P₀e^(rt))

Question 19

What is threshold population?

Accepted Answer

The smallest population a species needs in order to survive

Question 20

When is a first-order differential equation called linear?

Accepted Answer

When it can be written as a(x)y′ + b(x)y = c(x)

Question 21

Standard form of a linear first-order differential equation?

Accepted Answer

y′ + p(x)y = q(x)

Question 22

Integrating factor for y′ + p(x)y = q(x)?

Accepted Answer

μ(x) = e^(∫p(x) dx)

Calculus: Introduction to Differential Equations flashcards

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