Consider the differential equation y' = (y² + 3y + 2) ln(1 + y²). (a) Find the equilibria and determine their type. (b) Sketch the phase line. (c) Let y(t) be the solution satisfying y(0) = 1. Find the limit of y(t), as t→→∞.

(a) Consider the direction field of the differential equation dy/dx = x(y - 8)2 - 4, but do not use technology to obtain it. Describe the slopes of the lineal elements on the lines x = 0, y = 7, y = 8, and y = 9. X = 0 y = 7 y = 8 y = 9 (b) Consider the IVP dy/dx = x(y - 8)2 - 4, y(0) = Yor where y, < 8. Can a solution y(x) – o as x - o? Based on the information in part (a), discuss. O y(x) can approach o as x approaches o. O y(x) cannot approach co as x approaches o. Need Help? Read It Watch It

Given the differential equation = 2x (y² + 1) dy dx 5. (a) Sketch the slope field for this differential equation at the points shown in the figure. (-1,1) (0,1) (1,1) (-1,0) (0,0) (1,0) (-1,-1) (0,-1) (1,-1) (b) Let f be the particular solution to the differential equation whose graph passes through (0,1). Express f as a function of x, and state its domain.