Week 1 Jeopardy

Vocabulary

100

Define "Solution to a differential equation":

A solution of a differential equation is a differentiated function that satisfies the equation on some interval (a, b) of values for the independent variable.

200

dP/dt = kP[t]((1-P[t])/A) is an example of 

a) An initial condition

b) Invitial value problem

c) Logistic differential equations

Logistic differential equations

300

Define "Solution to a differential equation"

A solution of a differential equation is a differentiated function that satisfies the equation on some interval (a, b) of values for the independent variable.

400

What is "Order"?

The order of a differential equation is the order of the highest derivative present.

500

What's the difference between linear and non-linear differential equations?

A differential equation is linear in y if

-y and all its derivatives have pure functions od x 

-al the components are to the 1st power. 

Sketch Slope Fields

100

Sketch dy/dx=y

Check Homework solutions Week 1 Question 4

200

Sketch dy/dx=1-y²

Homework Solution for week 1 Question 5

300

Sketch dy/dt=y-t

Homework Solutions W01, question 6

400

Sketch dy/dt=y(y-1)

See Homework solutions W01, question 7

500

Sketch y'=1+3y+y²

Homework solution week1. question 8

Method of Separation (Differential equations)

100

Solve the following separable first-order differential equation

 xy'=1/y²

y=(4lnx+C)^1/4

200

Solve the following separable first-order differential equation

dy/dx=(x-5)/y²

y=((3x²/2)-15x+C)^1/3

300

Solve the following separable first-order differential equation

dy/dt=(-2ty)/t²

y=Kt^-2

400

Solve the following separable first-order differential equation

dx/dt=3xt³

x=Ke^{(3t^4)/4}

500

Solve the following separable first-order differential equation

x(dv/dx)=(1-4v²)/3v

V=((1-(kx)^-8/3)/4)^1/2

Vocabulary 2

100

Define "Initial Condition"

An initial condition for a differential equation gives an initial value

of the solution at some specific time t.

200

Define Initial Value Problem

An initial value problem is a differential equation of order n along with an initial condition.

300

What is a Slope field?

(or directional field) The slope field provided an intrusive way to understand a first-order differential equation where the slope at each point gives the direction of the solution and is shown by small vectors to denote the flow of solutions as time increases. In essence, the slope field shows all possible solutions. When an initial condition is used then the solution must navigate based on that initial point from where the function starts.

400

Define Equilibrium or constant solution

A solution to a DE is called an equilibrium or constant solution if

the function y is constant.

500

What is a Solution to a slope field?

A solution of a differential equation is a differentiated function that

satisfies the equation on some interval (a,b) of values for the independent variable.

Extra

100

Without using a calculator solve

The integral of (x²e^xdx)

x²e^x-2xe^x+2e^x+C

200

Integrate x²sin(x)dx

2Cos(x)-x²Cos(x)+2xSin(x)+C

300

Define if the following differential equation is linear or non-linear and explain why?

y'''+3yy"+y=e^x

Non-linear

There is not a pure function of x in front of y and its derivatives. 

400

Define if the following differential equation is linear or non-linear and explain why?

y''+e^xy'+y=Sinx

Linear

-Pure functions of x are accompanied by y and its derivatives.

y and its derivatives are to the 1st power. 

500

Name the name of the file and program to sketch Slope Fields on a computer

DFIELD Equation with Java