Solved: (a) The autonomous first-order differential equation dydx 1(y 3) has no critical

Chapter 2, Problem 40

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(a) The autonomous first-order differential equation dydx 1(y 3) has no critical points. Nevertheless, place 3 on the phase line and obtain a phase portrait of the equation. Compute d2ydx2 to determine where solution curves are concave up and where they are concave down (see 35 and 36 in Exercises 2.1). 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.