Differential Equations

Here is a fieldplot for a system of ODEs including a solution curve through a specified point.

> DEtools[DEplot]([diff(x(t),t)=-x(t)+3*y(t),diff(y(t),t)=2*x(t)+y(t)],[x(t),y(t)],t=-2..2,[[x(0)=2,y(0)=1]],x(t)=-6..6,y(t)=-6..6,linecolor=[blue],arrows=SMALL);

.......Remember that the user did not have to type in the Maple commands or custom Maple commands which are in red.......

[Maple Plot]

Here is the exact solution.

> F:=diffsolve({diff(x(t),t)=-x(t)+3*y(t),diff(y(t),t)=2*x(t)+y(t),x(0)=2,y(0)=1},{x(t),y(t)}): 

Maple found the exact solution.
Solutions for the the following dependent variables have been obtained.
            x            y
For example to access the solution for x use
  NameYouGaveResult[x]

> F[x];

[Maple Math]

> F[y];

[Maple Math]

Here we find the solution to the system by a classic Euler method.

> G:=diffsolve({diff(x(t),t)=-x(t)+3*y(t),diff(y(t),t)=2*x(t)+y(t),x(0)=2,y(0)=1},{x(t),y(t)},type=numeric,method=classical[foreuler],output=listprocedure): 

Maple found the solution numerically.  It has been returned as several procedures.
Solutions for the the following dependent variables have been obtained.
            y            x
For example to access the solution for y use
NameYouGaveResult[y](value of indepvar)
'NameYouGaveResult[y](indepvar)'
The use of the single quotes is required with table of values, limit table, and plot in MenuMaple.

.......With MenuMaple the user selects an action button for a Maple command or custom Maple command and enters required information after being prompted.........

This table allows us to compare the numerical solution for x (obtained by euler's method) with the exact solution for x.

> valutable(['G[x](t)',F[x]],t=-2..2,'step'=0.5); 

 t             funct1(t)        funct2(t)  
-----------------------------------------------
-2.           155.6142317      161.1079203     
-1.500000000  41.10635006      42.93490605     
-1.000000000  10.92754531      11.51486948     
-.500000000   3.161903258      3.361431313     
0             2.               2.              
.500000000    4.639261038      4.679708443     
1.000000000   16.52589834      16.81509480     
1.500000000   61.31027044      62.92655517     
2.000000000   228.1317280      236.1855173