Chapter 2.2, Problem 44E

1. (a) The autonomous first-order differential equation dy/dx = 1/(y − 3) has no critical points. Nevertheless, place 3 on the phase line and obtain a phase portrait of the equation. Compute d²y/dx² to determine where solution curves are concave up and where they are concave down (see Problems 35 and 36 in Exercises 2.1). Use the phase portrait and concavity to sketch, by hand, some typical solution curves.
2. (b) Find explicit solutions y₁(x), y₂(x), y₃(x), and y₄(x) of the differential equation in part (a) that satisfy, in turn, the initial conditions y₁(0) = 4, y₂(0) = 2, y₃(1) = 2, and y₄(−1) = 4. Graph each solution and compare with your sketches in part (a). Give the exact interval of definition for each solution.