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