Differential Equations

Homework 1
Differential Equations/Linear Algebra, Fall 2023 Due Friday, September 1 at 11 pm
Submission is online. See Canvas for submission instructions.

Please show your work and justify all answers.

  1. Evaluate the following linear combinations:
  2. Solve the following linear systems, and draw the row and column pictures corresponding to each system. (See Section 4.1 of the textbook for a reminder of what “row picture” and “column picture” 

x 3y = 2 3x + 2y = 4

2x y = 1

4x + 3y = 2

Find all solutions to the following linear systems, using elimination and back substitution. Label all the row operations that you perform:

2x + y z = 1 3x       z = 1

4x y + z = 0

6.

2x + 5y + z = 0 4x + 10y + z = 2

y z = 3

7.

2x 3y + 5z = 3 x + y 2z = 2 4x y + z = 7

8.

x y +  z +  t = 0

y −  z + 2t = 1 2x 3y + 3z                = 1

3x 4y + 4z + t = 1

For each of problems 5 through 8, write your answer v = y in the form v = vp + vn, where vp is a particular solution and vn is the family

of null solutions. (See page 205 of the textbook for an explanation of these terms.) Using matrix multiplication, verify directly that Avn = 0, where A is the matrix of coefficients for the system.

For which values of the parameter a does the following linear system have (i) a unique solution, (ii) no solution?

2x 3y = 0

x + ay = 1

For which values of the parameter a does the following linear system have (i) a unique solution, (ii) no solution, (iii) infinitely many solutions?

x + y + 7z = 7 2x + 3y + 17z = 16

x + 2y + (a2 + 1)z = 3a

(a) Find the unique quadratic polynomial y = ax2 +bx+c that passes through the three points (1, 1), (3, 5), and ( 2, 0) in the xy (Hint: this will involve a linear system where a, b, and c are the unknowns.)

(b) How many data points (x, y) should be needed to uniquely determine the coefficients of an nth degree polynomial anxn + an1xn1 + + a1x + a0? Explain your answer in terms of linear systems.

  1. Explain why a linear system Av = b cannot have exactly two solutions. (Hint: If the vectors v and w are both solutions, what is another solution?)
  2. Bonus question: You have three types of ground coffee: cheap ($5/pound), medium ($8/pound), and expensive Arabica ($14/pound). You want to make a blend that costs $10/pound, and you want to make 50 pounds of You also want to be able to say “contains at least 20% Arabica” without lying. Determine how much of each type of coffee you should use. (Hint: there is more than one possible correct answer.)