**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.

- Evaluate the following linear combinations:
- 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 **− *3*y *= 2 3*x* + 2*y* = 4

2*x **− **y *= *−*1

4*x *+ 3*y *= 2

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

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

4*x **− **y *+ *z *= 0

6.

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

*y **− **z *= 3

7.

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

8.

*x **− **y *+ *z *+ *t *= 0

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

3*x* *−* 4*y* + 4*z *+ *t* = *−*1

For each of problems 5 through 8, write your answer **v **= *y *in the form **v **= **v*** _{p}* +

**v**

*, where*

_{n}**v**

*is a particular solution and*

_{p}**v**

*is the family*

_{n}of null solutions. (See page 205 of the textbook for an explanation of these terms.) Using matrix multiplication, verify directly that *A***v*** _{n}* = 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?

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

*x *+ 2*y *+ (*a*^{2} + 1)*z *= 3*a*

(a) Find the unique quadratic polynomial *y *= *ax*^{2} +*bx*+*c *that passes through the three points (1*, *1), (3*, *5), and ( 2*, *0) in the *x*–*y * (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 *n*th degree polynomial *a*_{n}x* ^{n}* +

*a*

_{n}

_{−}_{1}

*x*

^{n}

^{−}^{1}+ +

*a*

_{1}

*x*+

*a*

_{0}? Explain your answer in terms of linear systems.

- Explain why a linear system
*A***v**=**b**cannot have exactly two solutions. (Hint: If the vectors**v**and**w**are both solutions, what is another solution?) **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.)