Differential Equations Jeopardy!!!!!

First-Order DEs

100

Method used to solve the DE: y' = M(x)N(y).

What is separation of variables?

200

x' + p(t)x = q(t)

What is a first-order linear differential equation?

300

A ___ occurs at the value of a parameter where the number of equilibrium solutions changes.

What is a bifurcation?

400

The solution to the IVP:

y' - (2/x)y = 2x², y(-2) = 4.

What is y(x) = 2x³+5x²?

500

The value of k in the DE: P' = kP, where P(0) = 1000 and P(1) = 1030. 

What is k = ln(1.03) = 0.0296?

Linear Systems

100

A nonzero vector, v, such that Av = Lv for some real # L

What is an eigenvector?

200

ad - bc

What is the determinant of the two by two matrix (a, b; c, d)?

300

Solution to the system:

x' = x + 3y

y' = x - y

x(t) = 3c₁e^2t + c₂e^-2t

y(t) = c₁e^2t - c₂e^-2t

400

Solution to the system x' = Ax, where A=(-1, -2; 4, 3).

What is vector_x(t) = c₁e^t(cos(2t); -cos(2t) + sin(2t)) + c₂e^t(sin(2t); -cos(2t) - sin(2t))?

500

The general solution of x' = Ax, where A = (4, 3; -3, -2).

What is x(t) = c₁e^t(1; -1) + c₂(t; -t + 1/3)e^t?

Geometric Analysis

100

Solutions to a differential equation y' = f(y) given by f(y) = 0 for all y.

What are equilibrium solutions?

200

An equilibrium solution such that all nearby solutions tend to converge towards it over time.

What is a sink?

300

The equilibrium points of the following IVP are classified as this:

x' = -(x/10) + y

y' = -x - (y/10)

x(0) = 2; y(0) = 2

What is a spiral sink?

400

The x-nullcline(s) of the system:

x' = x+y

y' = -2x+y

What is: y = -x?

500

The equilibrium points of the system:

x' = x(1-x-2y)

y' = y(1-y-2x)

What are: (0, 1); (1/3, 1/3); (1,0); (0,0)?

Second-Order DEs

100

a(t)x'' + b(t)x' + c(t)x = 0

What is a homogeneous second order DE?

200

Method used to solve this second-order DE:

x'' - 2x' -8x = e^-2t

What is undetermined coefficients?

300

y_p(t) = A_ntⁿ+A_n-1t^n-1+...A₁t+A₀

What is the guess for the particular solution if g(t) is an n^th degree polynomial?

400

The solution to:

y'' - y = cos(x)

What is y(x) = c₁e^x + c₂ - 0.5cos(x) - 0.5sin(x)?

500

The solution to the DE:

2y'' + 18y = 6tan(3t)

What is y(t) = c₁cos(3t) + c₂sin(3t) - (1/3)cos(3t)ln|sec(3t) + tan(3t)|?

Laplace Transforms

100

The variable that replaces t in a Laplace Transform

What is s?

200

F(s) = 2/((s-3)^2 + 4)

What is the Laplace Transform of f(t) = e^3tsin(2t)

300

F(s) = 1/(s(s+1))

What is the Laplace Transform of f(t) = 1 - e^-t?

400

f(t) = 3sin²(2t)

What is the inverse Laplace Transform of 24/s(s²+16)?

500

The solution to the IVP: y'' + 2y' + y = 2e^(-t), y(0)=-1, y'(0) = 3

What is y(t) = e^-t(t²+2t - 1)?