Consider the autonomous equation dp pp1 from problem 1 Solv

Consider the autonomous equation dp = p(p1) from problem 1. Solve this explicitly. Suppose that dt p(t) is a solution with p(0) = p0, where p0 > 1. Show that such a solution has a vertical asymptote at a finite time T (depending on p0), that is, there is an explosion at some T < . Find T explicitly in terms of p0. Use this information to again answer (d) in problem 1., this time with perhaps a slightly more accurate picture.

NOTE Q1 just to be able to answer the last part from the question above (

Consider the autonomous equation dp/dt = p(p 1).

(a) What are the equilibrium points of the equation?
(b) Draw a graph of p vs. dp . (Put p on the horizontal axis, dp on the vertical.)

(c) From your graph in (b), determine which of the equilibrium points are stable and which are unstable. Also determine where p is increasing and decreasing.

(d) Use this information to draw several trajectories, that is, draw a rough sketch of the graph of t vs. p(t) for several different values of the initial condition p(0). Include the equilibrium (i.e. constant) solutions in your sketch. )

Solution

a) equilibrium points are given by dp=0

so p(p-1)=0

p=0 or p-1=0

p=0,p=1 are equlibrium points