(*^
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Assignment #6
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RKStep[{t_,y_}] :=
( k1 = N[f[t,y]];
  k2 = N[f[t+0.5*h,y+0.5*h*k1]];
  k3 = N[f[t+0.5*h,y+0.5*h*k2]];
  k4 = N[f[t+h,y+h*k3]];
  N[{t+h,y+(h/6)*(k1+2*k2+2*k3+k4)}] )

:[font = input; preserveAspect; endGroup]
SetOptions[ListPlot,PlotJoined->True];
:[font = subsubsection; inactive; preserveAspect; startGroup]
1. y'[t] == 2 y[t] - 1, y[0] == 1
;[s]
2:0,0;3,1;34,-1;
2:1,13,9,Times,1,12,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect]
f[t_,y_] = 2*y - 1;
:[font = input; preserveAspect]
 h = 0.1;
;[s]
2:0,1;1,0;10,-1;
2:1,12,10,Courier,1,12,0,0,0;1,12,9,Times,0,12,0,0,0;
:[font = input; preserveAspect; startGroup]
solna=NestList[RKStep,{0,1},4]
:[font = output; output; inactive; preserveAspect; endGroup]
{{0, 1}, {0.1, 1.1107}, {0.2, 1.24590898}, 
 
  {0.3, 1.411053228172}, {0.4, 1.6127604128892808}}
;[o]
{{0, 1}, {0.1, 1.1107}, {0.2, 1.24591}, {0.3, 1.41105}, 
 
  {0.4, 1.61276}}
:[font = input; preserveAspect; startGroup]
pa = ListPlot[solna,PlotStyle->{RGBColor[0,1,0]}]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
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Graphics["<<>>"]
;[o]
-Graphics-
:[font = input; preserveAspect]
h = 0.05;
:[font = input; preserveAspect; startGroup]
solnb=NestList[RKStep,{0,1},8]
:[font = output; output; inactive; preserveAspect; endGroup]
{{0, 1}, {0.05, 1.052585416666666667}, 
 
  {0.1, 1.110701285425347222}, 
 
  {0.15, 1.174929248531268844}, 
 
  {0.2, 1.245912120040342831}, 
 
  {0.25, 1.324360319298419054}, 
 
  {0.3, 1.411058981045966534}, 
 
  {0.35, 1.506875813298388373}, 
 
  {0.4, 1.612769781646157627}}
;[o]
{{0, 1}, {0.05, 1.05259}, {0.1, 1.1107}, {0.15, 1.17493}, 
 
  {0.2, 1.24591}, {0.25, 1.32436}, {0.3, 1.41106}, 
 
  {0.35, 1.50688}, {0.4, 1.61277}}
:[font = input; preserveAspect; startGroup]
pb = ListPlot[solnb,PlotStyle->{RGBColor[0,0,1]}]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup]
%!
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[(1.4)] .01131 .39894 1 0 Msboxa
[(1.5)] .01131 .495 1 0 Msboxa
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Clear[t,y]
:[font = input; preserveAspect; startGroup]
soln = DSolve[{y'[t] == 2 y[t] - 1, y[0] == 1},
        y[t],t]
:[font = output; output; inactive; preserveAspect; endGroup]
{{y[t] -> 1/2 + E^(2*t)/2}}
;[o]
               2 t
          1   E
{{y[t] -> - + ----}}
          2    2
:[font = input; preserveAspect]
y[t_] = y[t] /. First[soln];
:[font = input; preserveAspect; startGroup]
{y[0.1],y[0.2],y[0.3],y[0.4]}
:[font = output; output; inactive; preserveAspect; endGroup]
{1.110701379080084917, 1.24591234882063516, 
 
  1.411059400195254487, 1.612770464246233802}
;[o]
{1.1107, 1.24591, 1.41106, 1.61277}
:[font = input; preserveAspect; startGroup]
p = Plot[y[t],{t,0,0.4}]
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