Unit Seven: Differential Equations- essential knowledge

What do differential equations relate?

Differential equations relate a function of an independent variable and the function’s derivatives.

How can derivatives be used in the context of differential equations

Derivatives can be used to verify that a function is a solution to a given differential equation

How many general solutions can there be to a differential equation?

There may be infinitely many general solutions to a differential equation

What is a slope field?

A slope field is a graphical representation of a differential equation on a finite set of points in the plane

What information do slope fields provide?

Slope fields provide information about the behavior of solutions to first-order differential equations.

What are solutions to differential equations?

Solutions to differential equations are functions or families of functions.

Which method can be used to solve some differential equations?

Some differential equations can be solved by separation of variables.

How can antidifferentiation be used in the context of differential equations?

Antidifferentiation can be used to find general solutions to differential equations.

What is the difference between a general solution and a particular solution to a differential equation?

A general solution may describe infinitely many solutions to a differential equation. There is only one particular solution passing through a given point.

What is a particular solution to the differential equation dy/dx=f(x) that satisfies F(a)=y₀

The function F defined by F(x)=y₀+∫axf(t)dt is a particular solution to the differential equation dy/dx=f(x), satisfying F(a)=y₀

Are solutions to differential equations always valid for all domains?

Solutions to differential equations may be subject to domain restrictions.

What are some specific applications of finding general and particular solutions to differential equations?

Specific applications include motion along a line and exponential growth and decay.

What is the model for exponential growth and decay that arises from the statement “The rate of change of a quantity is proportional to the size of the quantity

The model is dy/dx = ky

What is the solution to the exponential growth and decay model dy/dx = ky with initial condition y=y₀ when t=0

The solution is y=y₀e^kt