Exercises. Exercise 1: Write a method in Python and use it to estimate the solution of the initial value problem dy — 3 — 2t — 0.5у, у(0) — 1. dt (8) Plot the exact solution (9) y(t) = 14 – 4t –...

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Extracted text: Exercises. Exercise 1: Write a method in Python and use it to estimate the solution of the initial value problem dy — 3 — 2t — 0.5у, у(0) — 1. dt (8) Plot the exact solution (9) y(t) = 14 – 4t – 13e¬t/2 together with the estimated solution obtained via the tangent line method for step sizes h = 0.1, 0.2,0.5 in the interval [0, 1]. Exercise 2: Consider the differential equation dy (10) = y(a – y²) dt for the values a = -1, a = 0 and a 1 and determine their critical points. Sketch for each of these differen- tial equations their direction field and phase lines. Use these plots to determine whether the critical points are asymptotically stable or unstable.