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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 ๐0 COVID-10 viruses at time ๐ก = 0 and these viruses โmultipleโ subsequently according ๐โฒ(๐ก) = โ๐ผ๐(๐ โ ๐) where ๐(๐ก) is number of viruses at time ๐ก while ๐ผ 0 and ๐ 0 are constants. The patient will die when ๐ . If ๐0 ๐, the patientโs prognosis, i.e., the time ๐๐ is has left to live, is short. Derive a formula for ๐๐ and for given ๐0 = 1000, ๐ = 100, ๐ผ = 0.001. Compute the value of ๐๐.
Problem 3.2 (25 points): Find the PS of the following IVP:
๐ฆโฒโฒ โ ๐ฆโฒ โ 2๐ฆ = 0
{๐ฆ(0) = 1, ๐ฆโฒ(0) = 2
Problem 3.3 (25 points): Find the GS of the DE:
๐ฅ2๐ฆโฒโฒ โ 3๐ฅ๐ฆโฒ + 4๐ฆ = 0 Hint: substitute ๐ฅ = exp ๐ก.
Problem 3.4 (25 points): Solving the DE using the exact DE method:
4๐ฅ2 + 3๐ฆ2
๐ฆโฒ = โ
2๐ฅ๐ฆ