306D
Math Lab #3
Problem #1:
y
 t  c is an implicit solution of the logistic
1y
equation (2) y'  y(1  y) . This implicit solution can be derived by the method of separation of
variables (Can you derive it ?). To satisfy the initial condition y(0)  y0 we set
For any constant c the equation (1) F(y)  ln
y0
 (4) F(y)  F(y0 )  t . Equation (4) is the implicit solution of the initial
1  y0
value problem y'  y(1  y) , y(0)  y0 . More generally (5) F(y)  F(y0 )  t  t 0 is an implicit
solution of the initial value problem y'  y(1  y) , y(t 0 )  y0 .
(a) If we use the command "implicitplot" ( in the "plots" package) to sketch F(y)  F(y0 )  t  t 0
for a fixed value of y0 between 0 and 1 ( 0  y0 1 ) and for every value of t0 we generate the phase
portrait of the o.d.e. in the infinite, horizontal strip of the t-y plane between the constant solutions
y  0 and y  1 .
Let y0 .5 and use "implicitplot" to sketch F(y)  F(y0 )  t  t 0 for t0  2, 1,0,1,2 . You
should obtain
Figure 1:
(3) c  F(y0 )  ln
(b) If we use the command "implicitplot" to sketch F(y)  F(y0 )  t  t 0 for a fixed value of y0
greater than 1 (1  y0 ) and for every value of t0 we generate the phase portrait of the o.d.e. in the
infinite, horizontal strip of the t-y plane lying above the constant solutions y  1 .
Let y0  2 and use "implicitplot" to sketch F(y)  F(y0 )  t  t 0 for t0  2, 1,0,1,2 . You
should obtain
Figure 2:
(c) If we use the command "implicitplot" to sketch F(y)  F(y0 )  t  t 0 for a fixed value of y0
less than 0 ( 0  y0 ) and for every value of t0 we generate the phase portrait of the o.d.e. in the
infinite, horizontal strip of the t-y plane lying below the constant solutions y  0 .
Let y0  1 and use "implicitplot" to sketch F(y)  F(y0 )  t  t 0 for t0  2, 1,0,1,2 .
You should obtain
Figure 3
(d) Use the "display " command to combine the above graphs into a complete (global) phase portrait
of y'  y(1  y) . Be sure to include the constant solutions of y'  y(1  y) in your phase portrait.
Text #2= Differential Equations with Maple by Coombes, Hunt, Lipsman, Osborn, Stuck
C.O. Bloom
C.O. Bloom