Question

What is the solution for this first order nonlinear differential equation of this SIR model with these initial conditions? S(t)=not infected individuals (1) l(t)- Currently Infected (588) R(t)- recovered individuals (0) This will be a nonlinear first order differential equation(ODE) dasi d/dt-sal-kt di/dt a (s-k/a) i dr/dt-ki Total population will be modeled by this equation consistent with the SlR model. d(S+l+R)/dt= -saltsal-kltkl-0 Solution: i stk/aln stK Model the topic using a differential equation. a) Draw any visuals (diagrams) that exemplify the model and facilitate understanding of the modeling process. Your visuals should represent the topic and help develop the differential equation. b) Write the differential equation that models your topic of choice. Define all the variables used in the model, and be sure to include initial conditions. c) Classify the differential equation, including the order and if it is linear ornonlinear. 2 3 Solve the differential equation using the techniques from this course. a) Provide a valid solution to the differential equation. Be sure to apply the initial conditions that you developed. b) Discuss the method used to solve the differential equation. Explain all steps in arriving at the solution, writing in paragraph form and in complete sentences Discuss the solution in context of the initial topic. Write the solution of the differential equation in terms of the initial topic you chose in 4 Part 1. Describe how this could be useful for someone in the field from which the model was obtained. Include solution values or graphics to illustrate the utility and applicability of the solution in the context of the topic youchose. 5. Discuss what you learned from the modeling process. Write one to two paragraphs pertaining to what you can take away from the modeling process in general.

Answer 1

Let Tbe the temperature of the pot atanytimet giventhat T 100when t Omin T 80 whent = 5min T 65 when t = 10 min let T be the temperature of the kitchen. by newtonslawof cooling we have dT -k(T- dt _ dт -kat Т-7 log (T-) +c т-7, -е * T TAe 100 whent0min -t+c where A Ae _ giventhat T 100-7 A T T+(100-e* giventhat T 80 whent 5min, We have _ 80 7+(100-7e -5k 80-T -5k 100-T 80- -5k log 100-T 80- log 100-T 5 30- 100- we have T T +(100-7)e

giventhat T 65 when t = 10min 80- 100- 65 (100-7) (80- 2bg 100- 65 (100-) 80-T log 100- 65 70+(100-)e 80- 65 T+(100-0)100-T 65=7(80- 100-T 100 6400-1607 +7 65= 100-T 6500-657607+6400 5T 100 T20 0