Exam 4 Review Guide Blank

REVIEW GUIDE FOR EXAM 3, SYSTEMS AND LAPLACE TRANSFORMS SECTIONS 7.1 – 7.5, 6.1, 6.2 1. 1. If A =

1 1

b) Write Ax

0 . 2 3 1

a) How do we know that A–1 exists?

as a system of equations.

c) Find A–1 and use it to solve Ax

3 1

.

2 x1

2. a) Set up an augmented matrix for the system 2 x1

x2 x3

8 x4

6

4 x4

2 and completely reduce the 2

5 x4

augmented matrix using exactly two row operations.

b) Express the infinitely many solutions in vector form.

Review Guide for Exam 3, p. 1

1 0 3. Interpret the reduced matrix 0 1 0 0

4. Circle the vector, x (1 ) (t )

e 3t e 3t

1 2

0 0

0

1

or x (2 ) (t)

et , that is a solution of x' (t ) et

1 2 2 1

x.

5. Given the equation c1v(1) + c2v(2) + c3v(3) = 0. a) By definition the vectors v(1), v(2), and v(3) are linearly independent on some interval I if (circle one): i. ii.

This equation is true on I for nonzero values of c1, c2, or c3. This equation is true on I if and only if c1 = c2 = c3 = 0.

b) Find the relationship of dependency for the vectors: 1 0 2 v(1) =

2 , v(2) = 3

1 , v(3) = 1

0 2

Review Guide for Exam 3, p. 2

6. Convert the third order initial value problem, x' ' ' x' ' 4 x' 4 x 3t , x(0) = 1, x (0) = 2, x (0) = 3, into a corresponding system. Write the system in matrix-vector form.

7. The equation that defines r as an eigenvalue of the matrix A with corresponding eigenvector ξ is

8. Find the general solution of the system Dx

4 2

x . Express the general solution both as a linear 3 3 combination of solution vectors and in terms of (t).

Review Guide for Exam 3, p. 3

9. The eigenvalues of a third order system are r1

ξ

(1)

1 0 , ξ (2) 0

0 1 , ξ (3) 2

4 , r2

1 , r3 1 , with corresponding eigenvectors

0 5 . If the initial condition is x (0) 7

1 5 , find the specific solution. 1

10. Tanks A and B are interconnected. Tank A initially contains 15 lbs. of salt dissolved in 100 gallons of water. Tank B initially contains 100 gallons of pure water. Pure water flows into tank A at the rate of 3 gal/min. Brine flows from tank A to tank B at a rate of 4 gal/min. Brine flows from tank B to tank A at a rate of 1 gal/min. Brine leaves the system from tank B at a rate of 5 gal/min. Let x1(t) and x2(t) denote the amounts of salt in tanks A and B, respectively, at any time t. Set up a system of equations and initial vector to model this system.

Review Guide for Exam 3, p. 4

x1 ' 3 x1

x2

x3

11. Find the general solution of the system x 2 ' x1

x2

x3

x3 ' x1

x2

x3

Review Guide for Exam 3, p. 5

dI V (t ) . If R = 2, L = 1, and dt V(t) = e 2t , use Laplace transforms to find the zero-state output of the circuit.

12. The current flow of a circuit can be modeled by the equation RI

L

13. Use Laplace transforms to find the zero-state output of the mass-damper system modeled by u ' ' 2u ' u t 3 e t .

Review Guide for Exam 3, p. 6

14. Find the Laplace or inverse Laplace transforms as indicated. a) L (e 3t cos 2t )

b) L

1

c) L 1

s

3

s

2

2 2s 2

s

s 4s 5

Review Guide for Exam 3, p. 7