(a) The autonomous first-order differential equation dy/dx = 1/(y - 3) has no critic...

(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.

(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.

(reference problem 35 of exercise 2.1)

Using the autonomous equation (2), discuss how it is possible to obtain information about the location of points of inflection of a solution curve.

(reference problem 36 of exercise 2.1)

Consider the autonomous DE Use your ideas from Problem 35 to find intervals on the y-axis for which solution curves are concave up and intervals for which solution curves are concave down. Discuss why each solution curve of an initial-value problem of the form where -2 < y₀ < 3, has a point of inflection with the same y coordinate. What is that y-coordinate? Carefully sketch the solution curve for which y(0)=-1. Repeat for y(2) = 2

(reference problem 35)

Using the autonomous equation (2), discuss how it is possible to obtain information about the location of points of inflection of a solution curve.