Symbolic answers must be entered using notation similar to the examples given below. If you enter decimal numbers for questions that indicate that a symbolic answer is required, it will be marked as zero, regardless of whether the decimal number is correct or not. Integer numbers (positive, negative, and O) are allowed. Below are some examples. Example Math Expression How to Enter the Answer 2 -3/5 infinity 2\"8+l 2+exp(5) 5*pi/9 4/7-3*ln (2) sqrt(2) 5"(l/3) sqrt(5exp(3)+2*ln(7*pi)) fac(5) Problem #1: Solve the following initial value problem. ' = -10y1 + 312 -18y1 + 5y2 y'1(0) = 2, 32(0) = 3. Enter the functions y1(x) and y2(x) (in that order) into the answer box below, separated with a comma. Do not include 'y1 (x) = or 'y2(x) = in your answer. Enter your answer as a symbolic Problem #1: function of x, as in these examples Just Save Submit Problem #1 for Grading Problem #1 Attempt # 1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #2: In tracking the propagation of a disease, a population can be divided into 3 groups: the portion that is susceptible, S(t), the portion that is infected, F(t), and the portion that is recovering, R(t). Each of these will change according to a differential equation: S - WIT R' WITH so that the portion of the population that is infected is increasing in proportion to the number of susceptible people that contract the disease, and decreasing as a proportion of the infected people who recover. If we introduce the vector y = [S F R]', this can be written in matrix form as y' = Ay. If one of the solutions is y = x1 + 300e "/a x2 + 500e "C X3, where x1 = [0 0 50,000] , x2 = [0 -1 1]', and x3 = [6 15 -25]', what are the values of a, b, and c? Enter the values of a, b, and c into the answer box below, separated with commas. Problem #2:Problem # 3: A vector y = [R(t) F (1)]T describes the populations of some rabbits R(t) and foxes F(t). The populations obey the system of differential equations given by y' = Ay where A= 267 3510 -21 276 The rabbit population begins at 35100. If we want the rabbit population to grow as a simple exponential of the form R(t) = Roe\" with no other terms, how many foxes are needed at time t = 0? (Note that the eigenvalues of A are /l = 3 and 6.) I Just Save I I Submit Problem #3 for Grading I Problem #3 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #4: Which of the following systems could be used to obtain the solution to the differential equation yHI_ 5y"+7y7_4y = 0? (A) yf = y2 (B) y1' = y3 (C)y1' = y3 (D)y1' 2 V3 y2'=y3 y2'=y1 y2'=y1 y2'=y1 y3'=5y1+4y27y3 y3'=5y17y2+4y3 y3'=4y1'7J'2+5y3 y=4y1+5y2_7y3 (E) y; = y2 (F3 y; = y3 (G) y1' = y2 (H) yl' = V2 y2'=y3 y7f=y1 y2'=Y3 y=y3 y=4y1+5y2-7y3 y=5y1+4y2-7y3 y3'=5y1'7y2+4Y3 y=4y1_7y2+5y3 Problem #4: I Just Save I I Submit Problem #4 for Grading Problem #5: (a) Express the complex number (-2 +51) in the form a + bi. (b) Express the below complex number in the form a + bi. 4+ 3i i (4 + 6i) (c) Consider the following matrix. Let B = A". Find by1 (i.e., find the entry in row 1, column 1 of A-1) Problem #5(a): if your answer is a + bi, then enter a,b in the answer box Enter your answer symbolically, if your answer is a + bi, then enter a,b in the answer Problem #5(b): as in these examples box Enter your answer symbolically, if your answer is a + bi, then enter a,b in the answer Problem #5(c): as in these examples box Just Save Submit Problem #5 for Grading Problem #5 Attempt #1 Attempt #2 Attempt #3 Your Answer: | 5(a) 5 (a 5(a) 5 (b) 5 (b) 5 (b) 5 (c ) 5 (c) 5(c) Your Mark: 5(a) 5(a) 5 (a) 5 (b) 5 (b ) 5(b) 5(c) 5 (c ) 5 (c) Problem #6: Which of the following is a solution to the equation z' = (V3 + i)? (A) 21/3[cos(7/9) + i sin(7/9)] (B) 21/3[cos(8x/9) + i sin(8x/9)] (C) 213[cos(1 1/9) + i sin(11x/9)] (D) 2 3[cos(23x/18) + i sin(237/18)] (E) 2 3[cos(10x/9) + i sin(10/9)] (F) 21/3[cos(17/18) + i sin(17x/18)] (G) 21/3[cos(13/18) + i sin(13/18)] (H) 2 3[cos(19/18) + i sin(19x/18)] Problem #6: Select vProblem #7: gi @247 Find the value of 247 P bl #7. :| Enter your answer symbolically, if your answer is a + bi, then enter a,b in the answer ro em ' as in these examples box ' Just Save I ' Submit Problem #7 for Grading Problem #7 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #8: A system of differential equations can be created for two masses connected by springs between one another, and connected to opposing walls. The dependent variables form a 4 X 1 vector y consisting of the displacement and velocity of each of the two masses. For the system y' = Ay, the matriXA is given by: Because the system oscillates, there will be complex eigenvalues. Find the eigenvalue associated with the following eigenvector. 51' 51' 10 + 101' 10 101' Problem #8: |:| if your answer is a + bi, then enter a,b in the answer box Problem #9: Find the values of c1, C2, and c3 so that c1 (-10, -25, -3) + c2 (15, -5, 0) + c3 (5, 0, 0) = (-5, 10, 6). Problem #9: enter the values of C1, C2, and C3, separated by commas Just Save Submit Problem #9 for Grading Problem #9 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark