logistic2x.html

THE LOGISTIC EQUATION

The Equally-spaced Three Point Fit

MAPLE ALGEBRA NOTATION USED
eq2 := y = x^2	eq2 is the name of the equation y = x^2 naming the equations allows them to be referenced by name rather than formula
y = x^2	an equation
lhs(y=3*x+8)	left hand side of equation y=3*x+8
rhs(eq2)	right hand side of equation named eq2
numer( x^2/y)	numerator of fraction x^2/y
denom(frac4)	denominator of fraction frac4
solve(eq2,x)	solve equation eq2 for x
subs(x=3, y=t^2,eq2)	substitute x=3 and y=t^2 into equation eq2
collect(eq9,Q)	collect like terms in eq9 using Q as basis for determining what constitutes like terms
%	the output of the previous command
%[2]	the second output of the previous command if there was more than 1 output

> restart;

> `Logistic Equation`;eq1:= y = M/((1+((M-Y[0])/Y[0])*exp(-a*t)));

> `Solve eq1 for M`;eq2:= M = solve(eq1,M);

> `Substitute into eq2`;y = Y[1];t=h;eq3:= subs(t=h,y=Y[1],eq2);

> `Substitute into eq3`;y = Y[2];t=2*h;eq4:= subs(t=2*h,y=Y[2],eq2);

> `Set these two expressions eq3 and eq4 for M equal to each other`;eq5:= rhs(eq3)=rhs(eq4);

> `In eq5 substitute`;Q= exp(-a*h);eq6:= subs(exp(-a*h)=Q,exp(-2*a*h)=Q^2,eq5);

> `Cross multiply eq6 `;eq7:= numer(lhs(eq6))*denom(rhs(eq6)) = denom(lhs(eq6))*numer(rhs(eq6));

> `Transpose all terms to left in eq7`;eq8:= lhs(eq7)-rhs(eq7)=0;

> `In eq8 divide out common factor`; Y[0];`and simplify`;eq9:=simplify(eq8/(Y[0]));

> `In eq9 collect like terms in Q`;eq10:=collect(eq9,Q);

> `Solve quadratic equation eq10 for Q`; solve(eq10,Q);

> eq11:=Q = %[2];

> `Into eq11 substitute`; Q = exp(-a*h); eq12:= subs(Q=exp(-a*h),eq11);

> `Solve eq12 for a`;eq13:= a = solve(eq12,a);

> eq14:= Y[0] = Y[0];

> eq13;eq3;eq14;

This gives us the formulas for a, M and y0 in terms of y0, y1 and y2.