Maple Text Commands for Chapter 1 -  

DIFFERENTIAL EQUATIONS 

>	restart:with(DEtools):with(plots):

 

Example 1: Solve the differential equation  

>	deqn1:=diff(y(x),x)=-x/y(x): dsolve(deqn1,y(x));

 

(1)

 

>	C1:=x^2+y^2:

 

Figure 1.1: Plot solution curves for Example 1. 

>	implicitplot({C1=1,C1=4,C1=9},x=-4..4,y=-4..4,numpoints=1000,color=blue,scaling=CONSTRAINED,font=[TIMES,ROMAN,15],labels=[`t`,`x`]);

 

Example 3: The logistic equation modelling population growth. 

>	Logistic:=diff(P(t),t)=P(t)*(b-d*P(t)): dsolve(Logistic,P(t));

 

(2)

 

Example 8: The chemical reaction between nitrous oxide and oxygen to form nitrogen dioxide. Initial value problem  

>	k:=0.00713:a0:=4:b0:=1: deqn2:=diff(c(t),t)=k*(a0-(c(t))^2)*(b0-c(t)/2);

 

(3)

 

Figure 1.6: Plot a solution curve. 

>	DEplot(deqn2,c(t),t=0..1000,[[c(0)=0]],c=0..2.2,thickness=2,linecolor=blue);

 

Example 10: Solve the differential equation     

>	ode:=diff(i(t),t,t)+5*diff(i(t),t)+6*i(t)=10*sin(t): dsolve({ode,i(0)=0,D(i)(0)=0});

 

(4)

 

Exercise 7: Solve the system  

>	sys1:=diff(x(t),t)=-a*x(t),diff(y(t),t)=a*x(t)-b*y(t),diff(z(t),t)=b*y(t);

 

(5)

 

>	dsolve([sys1,x(0)=M,y(0)=0,z(0)=0]);

 

(6)

 

Example: Plot a solution curve for  

>	sys1:=diff(x(t),t)=z(t)-x(t),diff(y(t),t)=-y(t),diff(z(t),t)=z(t)-17*x(t)+16: DEplot3d({sys1},{x(t),y(t),z(t)},t=0..10,[[x(0)=0.8,y(0)=0.8,z(0)=0.8]],scene=[x(t),y(t),z(t)],stepsize=0.05,linecolor=blue,thickness=1);

 

Example: A time series plot. 

>	DEplot({sys1},{x(t),y(t),z(t)},t=0..10,[[x(0)=0.8,y(0)=0.8,z(0)=0.8]],scene=[t,x(t)],stepsize=0.05,linecolor=blue,thickness=1);

 

Example 7: Solve the initial value problem  

>	LDE:=diff(x(t),t)+t*x(t)=t^3;

 

(7)

 

>	dsolve({LDE,x(0)=1},x(t));

 

(8)

 

>	seriessolLDE:=dsolve({LDE,x(0)=1},x(t),series);

 

(9)

 

Example 8: Solve the van der Pol differential equation  

using numerical methods and a series solution. Compare the results near to the initial conditions. 

>

restart:with(DEtools):with(plots): vanderPol:=diff(x(t),t,t)+2*(x(t)^2-1)*diff(x(t),t)+x(t)=0; numsol:=dsolve({vanderPol,x(0)=5,D(x)(0)=0},numeric,range=0..0.08): p1:=odeplot(numsol): seriessol:=dsolve({vanderPol,x(0)=5,D(x)(0)=0},x(t),series); approxpoly:=convert(rhs(seriessol),polynom); p2:=plot({approxpoly},t=0..0.08,colour=blue): display({p1,p2},font=[TIMES,ROMAN,15],title=["Numerical and Series Solutions"]);

 

>

 

End of Chapter 1 Commands