Jan Zijlstra, Math 3120 Fall 2006

Section A.1  A Learning Model

Model Hypothesis:	'The rate of learning is proportional to the amount left to learn'
Corresponding DE:	y'(t) = k ( 100 - y(t) )
Symbols:	y(t): amount mastered, in % of the learning task, at time t (hours). y'(t): rate of learning, in % of learning task per hour, at time t.   k: rate constant.

> eq1 := diff(y(t),t)=k*(100-y(t));

> sol1A := dsolve(eq1,y(t));

> init1 := y(0)=20;

> sol1B := rhs(dsolve({eq1, init1}, y(t)));

> plot(subs(k=0.05, sol1B),t=0 . . 1,color=blue,
     title=`solution of the learning model for k=0.05`);

> with(plots):
    animate(sol1B,t=0..1,k=0..0.1,color=blue,
    title=`Effect of the value of k on the solution of the learning model`);

Section A.2.  A Model for Exponential Growth

Model Hypothesis:	'The population growth is proportional to the population size '
Corresponding DE:	y'(t) = k y(t)
Symbols:	y(t) : population size, in millions, at time t (years). y'(t) : population growth, in millions per year, at time t.   k: rate constant.

> eq2 := diff(y(t),t)=k*y(t);

> sol2A := dsolve(eq2,y(t));

> init2 := y(0)=3/2;

> sol2B := rhs(dsolve({eq2} union {init2}, y(t)));

> plot(subs(k=0.05,sol2B),t=0..10,color=blue, title=`solution of the exponential problem for k=0.05`);

> animate(sol2B,t=0..1,k=0..0.1,color=blue,
     title = `Effect of k-value on solution of the exponential problem`);

Section A.3.  A Logistic Growth Model

Model Hypothesis:	'The relative population growth is proportional to the difference between  the maximum sustainable population, M ,and the population size'
Corresponding DE:	y'(t)/y(t)= k [M - y(t)]  or  y'(t) = k y(t) [M - y(t)]
Symbols:	y(t) : population size, in millions, at time t (years) y'(t)/y(t) : relative population growth, in millions per million per year, at time t. k : rate constant.

> eq3 := diff(y(t),t)=k*y(t)*(M-y(t));
   sol3A := dsolve(eq3,y(t));

> init3:=y(0)=3;
    sol3B := rhs(dsolve({eq3} union {init3}, y(t)));

> plot(subs(M=200,k=0.05,sol3B),t=0..1,color=blue,
    title = `solution of logistic problem for M=200 and k=0.05`);

> animate(subs(M=200,sol3B),t=0..1,k=0..0.5,color=blue,
     title = `Effect of k-value on solution of logistic problem`);

> animate(subs(k=0.05,sol3B),t=0..1,M=0..200,color=blue,
     title = `Effect of M-value on solution of logistic problem`);