-- Euler routine which solves simultaneous differential equations.
-- Results appear at very bottom of this Pad

-- by Anthony Egan B.S.
-- SD School of Mines ChE
-- afe8607@silver.sdsmt.edu

Euler(x,y)[i] = y[i] + dxx * ydot(x,y)[i]


iters:=100:;   -- the number of iterations

ny = 3         -- the number of equations

--input an array of ny differential equations

ydot(x,y)={ y[1],
            y[1]-y[2],
            x^2-x }

------
zeros2[i,j] = 0 dim[ny,iters]
xx:=0[:iters]:
y:= zeros2:

--input initial conditions of the dependent variables

y[1,1]:=1:
y[2,1]:=4:
y[3,1]:=1:

--input the initial and final values for independent variable x

xinitial:=2:
xfinal:=6:

--Begin Program

dxx:=(xfinal-xinitial)/iters:; dxx:0.040
xx[1]:=xinitial:

i:=1:
(
 k:=1,
 (y[k,i+1]:= Euler(xx[i],y)[k,i],k:=k+1) while k<=ny,
  xx[i+1]:=xx[i]+dxx,i:=i+1) while i<iters:

--Begin Final Results

y[1,iters]:48.562
y[2,iters]:24.343
y[3,iters]:52.592

--plot of results.
Title:="y's vs x":;
Xmin:=xinitial:;  Xmax:=xfinal:
        Ylabel:=" y[1]":; plotline {xx,y[1]}
newaxis;Ylabel:=" y[2]":; plotline {xx,y[2]}
newaxis;Ylabel:=" y[3]":; plotline {xx,y[3]}

--table of results------------------------
increment:=5:
Nmin:=1:
Nmax:=iters:
Nsteps:=iters/increment:
table xx[N],    y[1,N],    y[2,N],    y[3,N]
      2.000      1.000      4.000      1.000
      2.200      1.217      3.462      1.450
      2.400      1.480      3.067      2.034
      2.640      1.873      2.758      2.935
      2.840      2.279      2.625      3.872
      3.040      2.772      2.597      4.994
      3.240      3.373      2.674      6.317
      3.440      4.104      2.857      7.858
      3.680      5.193      3.227     10.017
      3.880      6.318      3.673     12.094
      4.080      7.687      4.262     14.438
      4.280      9.352      5.018     17.068
      4.520     11.833      6.184     20.622
      4.720     14.397      7.416     23.935
      4.920     17.516      8.936     27.584
      5.120     21.311     10.800     31.584
      5.320     25.928     13.082     35.953
      5.560     32.807     16.496     41.704
      5.760     39.915     20.033     46.938
      5.960     48.562     24.343     52.592