Visualizing Differential Equations

Visualizing Differential Equations

• 1.
  PEI DU
• 2.
  The Gompertz equationhas been used to model self-limited population growth. It is characterized by the differential equation dy/dt=r*y*ln(K/y) along with the initial condition y(0)=y0. Since it is widely used in science and is a typical example for ordinary differential equations, I want to use these slides to show how to draw its graphs using Maple.
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• 4.
  graph Maple codeswith(DEtools): Phaseportrait(D(y)(t))=2*y(t)*ln(3/y(t)), y(t), t=1..6,[[y(0)=0.5],[y(0)=1],[y(0)=1.5]],title=‘Asymptotic solution’, colour=magenta,linecolor=[gold, yellow, wheat]);
• 5.
  Maple codes: >with DEtools: > y: =‘y’: > eqn: diff(y(t),t)=2y(t)ln(3/y(t)); > dfieldplot(eqn, y(t),t=1..6,y=-1..4)
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  with(plots): p1:=dsolve({D(y)(t)=2*y(t)*ln(3/y(t)),y(0)=1},y(t), type=numeric):p2:=dsolve({D(y)(t)=3*y(t)*ln(3/y(t)),y(0)=1},y(t), type=numeric): p3:=dsolve({D(y)(t)=4*y(t)*ln(3/y(t)),y(0)=1},y(t), type=numeric): p4:=dsolve({D(y)(t)=5*y(t)*ln(3/y(t)),y(0)=1},y(t), type=numeric): p5:=dsolve({D(y)(t)=6*y(t)*ln(3/y(t)),y(0)=1},y(t), type=numeric):
• 8.
  a1:= odeplot(p1,[t, y(t)],-1..3,color=blue): a2:= odeplot(p2,[t, y(t)],-.8..3, color=green): a3:= odeplot(p3,[t, y(t)],-.6..3, color=yellow): a4:= odeplot(p4,[t, y(t)],-.4..3, color=orange): a5:= odeplot(p5,[t, y(t)],-.4..3, color=red): Display(a1,a2,a3,a4,a5);
• 9.
  graph on MapleMaple codes With(plots): eqn: =D(y)(t)=C*y(t)*ln(3/y(t)): toplot={seq(subs(C=i, eqn), i=2..6)}; plot(toplot, t=-5..5,-5..5);
• 10.
  The graphing codesshowed above can be extended to many other ordinary differential equations. Maple has many handy graphing functions. It can be used to simple functions, direction fields, asymptotes, and families of functions. Maple also has functions to create animated functions.
• 11.
  Special thanks tomy tutor Kathy. : )