For the equation dy/dt = y4 + 4y3 − 5y2 (a) Find all the equilibrium solutions, and determine their stability type. (b) Sketch the phase line and draw several solution curves in the ty-plane. (c) Determine all the inflection levels, if any at all, for the solutions curves.

Consider the autonomous differential equation. y' = (y – 1)(3 – y)² Sketch solutions curves in the t – y plane corresponding to the following initial conditions. Make sure you label your graph and label any equilibrium solutions. y(0) = 4 y(0) = 3 y(0) = 2 y(0) = 1 y(0) = 0. %3D

Suppose y2 2 1 + cze? 1 ý(t) = y1 y1 -3 -3 -2 -1 5(1) = -2 -2 A B (a) Find c, and c2. y2 y2 2 C1 = 1/2e 1 y1 -e/2 -3 C2 = -1 -3 1 -1 -2 (b) Sketch the phase plane trajectory that satisfies the given initial condition. Which graph most closely resembles the graph you drew? D D (c) What is the approximate direction of travel for the solution curve, as t increases from -o to -+o? O A. along the line y = -x toward the origin and then along the line y = x away from the origin O B. along the line y = x toward the origin and then along the line y = -x away from the origin C. none of the above

5. Recall that a 1st order, autonomous equation can be written in general as 4 = f(y). Consider the equation: dy dr y3 – y² – 2y (a) Find the equilibrium solutions (critical points) of the equation. (b) Draw a gram of y vs. f(y) (make sure to label the ch). ase (c) Sketch of the actual solutions of æ vs. y (make sure to label the sketch). (d) Characterize the stability of the equilibrium solutions (e.g. stable, unstable, semi-stable).