Example 2: Logistic Equation

> de2:=diff(y(x),x) = 0.5*y(x) - 0.1*y(x)^2;

> L:=dfieldplot(de2,y(x),x=0..12,y=0..8,arrows=medium):

> display({L});

> dsolve({de2,y(0)=1},y(x));

> y0 := x -> 5.*1/(1.+4.000000000*exp(-.5000000000*x));

> dsolve({de2,y(0)=2},y(x));

> y1 := x -> 5.*1/(1.+1.500000000*exp(-.5000000000*x)):

> dsolve({de2,y(0)=8},y(x));

> y2 := x -> 5.*1/(1.-.3750000000*exp(-.5000000000*x)):

> M0:=plot(y0(x),x=0..12,colour=blue,thickness=3):

> M1:=plot(y1(x),x=0..12,colour=yellow,thickness=3):

> M2:=plot(y2(x),x=0..12,colour=blue,thickness=3):

> display({L,M0,M1,M2});