Math 2670 Exam 2 Jeopardy

Real Distinct Eigenvalues

100

Find a solution to the following system:

x' = x + 2y

y' = 2x + y

Ae^3t (1 1) + Be^-t (1 -1)

200

Find a solution to the following system

x' = -2x - y

y' = 2x - 5y

Ae^-3t (1 1) + Be^-4t (1 2)

300

Find a solution to the following system:

x' = x + y

y' = -2y + 4x

Ae^2t (1 1) + Be^-3t (1 -4)

Repeated Eigenvalues

100

Find the exponential matrix for the following system:

x' = 5x

y' = x + 5y

e^At = e^5t * (1 0, t 1)

200

Find the exponential matrix for the following system:

x' = 3x - y
y' = x + 5y

e^At = e^4t * (1-t  -t, t  1+t)

300

Find the exponential matrix for the following system:

x' = -x - y

y' = 4x - 5y

e^At = e^-3t * (1+2t  -t, 4t  1-2t)

Complex Eigenvalues

100

Find the exponential matrix for the following system:

x' = 2y

y' = -2x

e^At = (cos(2t)   sin(2t), -sin(2t)   cos(2t))

200

Find the exponential matrix for the following system:

x' = x - 2y

y' = x + 3y

e^At = e^2t * 

(cos(t) - sin(t)   -2sin(t), sin(t)   cos(t) + sin(t))

300

Find the exponential matrix for the following system:

x' = -x - 4y

y' = -y + x

e^At = e^-t * 

(cos(2t)   -2sin(2t), .5 sin(2t)  cos(2t))

Miscellaneous

100

What is this equation primarily used for in this course?

e^At=Φ(t) * (Φ(0))^(-1)

To find the exponential matrix given any fundamental matrix, but for this course it's mostly used when I want to find exponential matrix for systems with real distinct eigenvalues

200

What is the name for the method where I utilized to D operator to solve a system of differential equations?

The elimination method

300

How do I approach rewriting a higher order differential equation as a system of first order differential equations?

Substitute in variables, allowing each to represent a derivative of the previous, and the rewriting the original DE to solve for the last variable