[SOLVED] AMS361 Homework3

Problem 3.1 (25 points): An outbreak of COVID-19, a highly contagious disease, is ravaging through all over the world. We hypothesize: a patient is diagnosed to be have contracted 𝑃₀ COVID-10 viruses at time 𝑡 = 0 and these viruses “multiple” subsequently according 𝑃^′(𝑡) = −𝛼𝑃(𝑀 − 𝑃) where 𝑃(𝑡) is number of viruses at time 𝑡 while 𝛼 > 0 and 𝑀 > 0 are constants. The patient will die when 𝑃. If 𝑃₀ > 𝑀, the patient’s prognosis, i.e., the time 𝑇_𝑝 is has left to live, is short. Derive a formula for 𝑇_𝑝 and for given 𝑃₀ = 1000, 𝑀 = 100, 𝛼 = 0.001. Compute the value of 𝑇_𝑝.

Problem 3.2 (25 points): Find the PS of the following IVP:

𝑦^′′ − 𝑦^′ − 2𝑦 = 0

{𝑦(0) = 1, 𝑦′(0) = 2

Problem 3.3 (25 points): Find the GS of the DE:

𝑥²𝑦^′′ − 3𝑥𝑦^′ + 4𝑦 = 0 Hint: substitute 𝑥 = exp 𝑡.

Problem 3.4 (25 points): Solving the DE using the exact DE method:

4𝑥² + 3𝑦²

𝑦′ = −

2𝑥𝑦