> 
# Phase diagrams for 1st order ODEs are considered and studied.
#
--------------------------------------------------------------------------------
# 
# Art Belmonte
# Thu, 25/Jan/96
# Math 308-509
# Section 2.6: Population Dynamics and Some Related Problems
# 
# Recall that `T' stands for the textbook by  Boyce and Diprima (`M' 
# would signify lab manual), 22/1 means page 22 / problem #1, and 
# that S2.1 indicates the problem is from Section 2.1
--------------------------------------------------------------------------------
# T-62/3-1:
# dN/dt = F
> F:=N->N * (N-1) * (N-2);\
plot(F(N), N=0..3, y=-1..2, labels=[`N`, `dN/dt`],\
     xtickmarks=3, ytickmarks=2);
--------------------------------------------------------------------------------
# T-62/3-2: NOTE: The overflow errors/warnings are probably 
# indicative of an asympotote in the solution corresponding to the 
# initial condition N(0) = 2.1.
> unassign('N');\
deq:={diff(N(t), t) = N(t) * (N(t) - 1) * (N(t) - 2)};\
inits:={[0, 0.2], [0, 0.8], [0, 1.7], [0, 2.1]};\
DEplot1(op(deq), N(t), t=0..3, inits, N=-1..3);
--------------------------------------------------------------------------------
# T-62/3-3: Here are implicit and explicit solutions from dsolve. (It 
# this case dsolve COULD explicitly solve for N(t). This may not 
# always be the case.) Note that this NONLINEAR equation has 
# three equilibrium solutions which are NOT accounted for in the 
# dsolve solutions: N(t) = 0, N(t) = 1, and N(t) = 2. This is why we 
# can't speak of a "general" solution to a nonlinear differential 
# equation like we can for a linear one.
> deq; dsolve(deq, N(t));\
dsolve(deq, N(t), explicit);
--------------------------------------------------------------------------------
# T-63/9-1: 
> unassign('N');\
F:=N->N^2 * (N^2-1);\
plot(F(N), N=-2..2, y=-1..2, labels=[`N`, `dN/dt`],\
     xtickmarks=3, ytickmarks=2);\

--------------------------------------------------------------------------------
# T-63/9-2: NOTE: The overflow errors/warnings are probably 
# indicative of an asympotote in the solution corresponding to the 
# initial condition N(0) = 1.1.
> unassign('N');\
deq:={diff(N(t), t) = N(t)^2 * (N(t)^2 - 1)};\
inits:={[0, 0.2], [0, 0.8], [0, 1.1], [0, -0.3], [0, -1.7]};\
DEplot1(op(deq), N(t), t=0..3, inits, N=-2..2);
--------------------------------------------------------------------------------
# T-63/9-3: Here is an implicit solution to the D.E. Note that dsolve 
# was not able to isolate an explicit solution. Note that this 
# NONLINEAR equation has three equilibrium solutions which are 
# NOT accounted for in the dsolve solution: N(t) = -1, N(t) = 0, and 
# N(t) = 1. This is why we can't speak of a "general" solution to a 
# nonlinear differential equation like we can for a linear one.
> deq; dsolve(deq, N(t));\
dsolve(deq, N(t), explicit);
--------------------------------------------------------------------------------