AMS 361: Applied Calculus IV
Homework 3
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 𝑇𝑝 is has left to live, is short. Derive a formula for 𝑇𝑝 and for given 𝑃0 = 1000, 𝑀 = 100, 𝛼 = 0.001. Compute the value of 𝑇𝑝.
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
