(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. Use the phase portrait and concavity to sketch, by hand, some typical solution curves.

(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.