Supplementary Problems: Phase Portraits & Chemical Reactions in MA 532 Diff. Equations - P, Assignments of Differential Equations

Download Supplementary Problems: Phase Portraits & Chemical Reactions in MA 532 Diff. Equations - P and more Assignments Differential Equations in PDF only on Docsity! MA 532 Supplementary Problems 1 August 20, 2003 1. Sketch the phase portrait, the direction field, and some typical solutions. (a) ẋ = x− x3 (b) ẋ = 1 + x2 (c) ẋ = x2 − x3 (d) ẋ = ex sin x (e) ẋ = 1− 2 cosx 2. The velocity v(t) of a falling skydiver is governed by the differential equation mv̇ = mg − kv2, where m is the mass of the skydiver, g is the acceleration due to gravity, and k is a constant related to air resistance. The constants m, g, and k are positive. (a) Sketch the phase portrait. (b) What is the skydiver’s terminal velocity? 3. Consider the model chemical reaction A + X r1 → r2 ← 2X in which one molecule of X combines with one molecule of A to form two molecules of X. This means that the chemical X stimulates its own production, a process called autocatalysis. This positive feedback process is eventually limited by a “back reaction” in which 2X returns to A+X. According to the law of mass action of chemical kinetics, the rate of an elementary reaction is proportional to the product of the concentrations of the reactants. We denote the concentrations of A and X by a and x respectively. Assume that there is an enormous surplus of chemical A, so that its concentration a can be regarded as constant. Then the differential equation for this reaction is ẋ = r1ax− r2x 2 where r1 and r2 are positive rate constants. 1