Updated 2013

2.1-8:   Solve y'=7y(y-13) with y(0)=17
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 Step 1. Identify as variables separable y'=F(x)G(y), with F=7,
         G=y(y-13). 
 Step 2. Divide by G(y) and apply quadrature to get 7x+c on the
         right and a partial fraction integration problem on the
         left:

           |   y'dx  |
        int|---------| = 7x+c
           | y(y-13) |
                             1        A       B
 Step 3. Partial fractions ------- = ---  + ------
                           y(y-13)    y     y - 13

        Use a sampling method to solve for A=-/13, B=1/13.  

 Step 4. Evaluate the constant c using y(0)=17.
 Step 5. Solve for y(x), the explicit solution. Use log and
         exponential rules and the trick |u| = +1 or -1 times u.
 Step 6. Answer check. The book answer is correct. The text also has
         a phase diagram drawn, which matches what you could do in
         two minutes from the phase line diagram method in class.
         See textbook section 2.2 for more info.

 MAPLE ANSWER CHECK 
  # y'=7y(y-13) with y(0)=17
  f:=(x,y)->7*y*(y-13);
  de:=diff(y(x),x)=f(x,y(x)); # Symbol y(x), not y!
  ic:=y(0)=7;
  sol:=dsolve([de,ic],y(x));
  ans:=rhs(sol);
  plot(ans,x=0..0.5,y=0..1);
  # answer  91/(7+6*exp(91*x))  

2.1-16: Solve P'=aP-bP^2, given P(0)=120 and aP=8, bP^2=6 at t=0.
=======

 The rabbit problem. Information in English in the problem statement
 implies P'=aP - bP^2, P(0)=120. The terms aP and bP^2, known as
 birth and death terms, are given at t=0: aP(0)=8, b(P(0))^2=6.
 Insert P(0)=120 to find a=1/15 and b=6/(120)^2. Let M=a/b=carrying
 capacity. Then M=160. Use the book formula in section 2.1 for P(t):

                      M P(0)
        P(t)= ----------------------
              P(0)+(M-P(0))exp(-a t)

 Fill in all the constants and then solve P(T)=0.95 M for T=15
 ln(95/15).

                      M P(0)
        P(T)= ----------------------
              P(0)+(M-P(0))exp(-a T)

        95M           M P(0)
        --- =  ----------------------     Cancel M 
        100    P(0)+(M-P(0))exp(-a T)
       
        95             120
        --- =  ----------------------     Subst P(0)=120, 
        100    120+(160-120)exp(-T/15)    a=1/15, M=160     
       
 To Finish: Cross-multiply and isolate exp(-T/15) on the left. Take
 the log of both sides, simplify, and find the value of unknown T.

 MAPLE ANSWER CHECK 
 # P'=aP-bP^2 with a=1/15, b=6/120^2, P(0)=120
   a:=1/15:b:=6/120^2:
   f:=(t,P)->a*P-b*P^2;
   de:=diff(P(t),t)=f(t,P(t)); # Symbol P(t), not P!
   ic:=P(0)=120;         
   sol:=dsolve([de,ic],P(t));
   ans:=rhs(sol);
   plot(ans,t=0..100);
   # ans := 480/(3+exp(-t/15))