In Problems 43 and 44 we saw that every autonomous first-order differential equation dy/...

In Problems 43 and 44 we saw that every autonomous first-order differential equation dy/dx = f (y) is separable. Does this fact help in the solution of the initial-value problem Discuss. Sketch, by hand, a plausible solution curve of the problem.

(reference problem43 )

Every autonomous first-order equation dy/dx = f (y) is separable. Find explicit solutions y₁(x), y₂(x), y₃(x), and y4(x) of the differential equation dy/dx = y - y³ that satisfy, in turn, the initial conditions y₁(0) = 2, and y₄(0)=-2. Use a graphing utility to plot the graphs of each solution. Compare these graphs with those predicted in Problem 19 of Exercises 2.1. Give the exact interval of definition for each solution.

(reference problem 19 of exercise 2.1)

Consider the autonomous first-order differential equation dy/dx = y - y³ and the initial condition y(0) = y₀. By hand, sketch the graph of a typical solution y(x) when y0 has the given values.

(reference problem 44)

(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.