By John Earman
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Extra resources for A primer on determinism
1 Core Exercises The goal of the Core exercises is as follows: To deal with unﬁnished business To practice solving separable ODEs To increase your general facility with ODEs 1. (a) 1–3 4 5–7 Show that every solution of x = ax equals Ceat for some constant C. Hint: Show that for a solution x(t), the derivative of e−at x is zero. 17). Hint: Write ea¯(s)−¯a(t) = ea¯(s) e−¯a(t) and move e−¯a(t) outside the integrand. Then diﬀerentiate using the product rule. (c) Prove the following claims about the logistic equation made in the text.
Friction of a mass sliding over a dry surface. Despite this inaccuracy, friction is widely approximated by such a term, because of the appealing fact that this leads to linear ODEs. Combining the above forces in Newton’s equation, we get the ODE for the motion of the mass mx = −bx − kx. 24) is a linear combination of exponentials (apart from the exceptional 6 You may groan that we use the same letter b for the constant of proportionality as for the initial conditions. Please trust us—in the larger scheme of things, this conﬂict of notation will not cause confusion.
18 Chapter 1. 6: An inductor, a resistor, and a capacitor in parallel. 34). 33). 35) where the parameters a, b, c, d are all positive. 35), since in our view, such understanding is an essential part of acquiring facility with ODEs. This system describes the evolution of two interacting populations, a predator (say sharks, represented by y) and a prey (say regular ﬁsh or teleosts,16 represented by x). , y = 0), the prey population satisﬁes x = ax, the equation for exponential growth. 17 Similarly, the predator equation for y represents a balance between two eﬀects: the predator population increases by a term proportional to the amount of food the predators consume and decreases by a “death” term proportional to their population.