The same disease is spreading through two populations, say Pi and P2, with the same size. You may assume that the spread of the disease is well described by the SIR model. dSa - BaSaIa dt dla - dt dRa YaIa dt with Sa(0) + I.(0) + R.(0) = N where N denotes the fixed population size. The subscript a identifies the population P or P2. For example, if a = 1, the variables are related to P1. Assume that S1 (0) = S2 (0) and I1(0) = I½(0) and that no interventions such as quarantine or vaccination have been implemented. If the difference in the spread of the disease is due only to the poor over-all health of a population, which population has the best over-all health of the two populations? - Susceptible Population 1 - Susceptible Population 2 700 600 500 400 300 200 100 10 15 20 t P1. P2. Susceptible

1. Describe and solve each first-order equation. (a) x²y'=xy²+3y² (b) y'+3y=3t²e-³t

6.2.10. Let us think of the mass-spring system with a rocket from Example 6.2.2. We noticed that the solution kept oscillating after the rocket stopped running. The amplitude of the oscillation depends on the time that the rocket was fired (for 4 seconds in the example). a) Find a formula for the amplitude of the resulting oscillation in terms of the amount of time the rocket is fired. b) Is there a nonzero time (if so what is it?) for which the rocket fires and the resulting oscillation has amplitude 0 (the mass is not moving)?

Do question number 3

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2.6-2. Given the difference equation - 2) = 3(k+ 1) + x(k)= y(k + 2) where y(0) = y(1) = 0, e(0) = 0, and e(k) = 1. k=1,2,... (a) Solve for y(k) as a function of k, and give the numerical values of y(k), (b) Solve the difference equation directly for y(k), 0 k 4. to verify (c) Repeat parts (a) and (b) for e(k)= 0 for all k, and y(0) = 1, y(1) = -2. the results 0 sks 4. of part (a). y (k)

Problem 4. Let 3 5 3 17 t 20 17 4 A = 1 1 1 8. 1 2 3 3 where t is a variable. (a) (Ъ) (c) Explain why det A = at? + bt + c, where a, b, and c are constants. Find a. Find det A when t = 0. Also, find det A when t = 4. No calculation is required. Use this to find b and c.

Problem 5. Let A and k be positive constants. Which of the given functions is a solution to = k(y + A)? O A. y = A-1 + CeAkt O B. y = -A + Cekt O C. y = -A+ Ce-kt O D. y = A+ Cekt O E. y = A+ Ce-kt OF. y = A-+ Ce-Akt