in mathematica, how to make initial condition as a variable in ndsolve? - wolfram-mathematica

Question

i'd like to have something like this
w[w1_] :=
NDSolve[{y''[x] + y[x] == 2, y[0] == w1, y'[0] == 0}, y, {x, 0, 30}]
this seems like it works better but i think i'm missing smtn
w := NDSolve[{y''[x] + y[x] == 2, y[0] == w1, y'[0] == 0},
y, {x, 0, 30}]
w2 = Table[y[x] /. w, {w1, 0.0, 1.0, 0.5}]
because when i try to make a table, it doesn't work:
Table[Evaluate[y[x] /. w2], {x, 10, 30, 10}]
i get an error:
ReplaceAll::reps: {<<1>>[x]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >>
ps: is there a better place to ask questions like that? mathematica doesn't have supported forums and only has mathGroup e-mail list. it would be nice if stackoverflow would have more specific mathematica tags like simplify, ndsolve, plot manipulation

Answer 1

There are a lot of ways to do that. One is:
w[w1_] := NDSolve[{y''[x] + y[x] == 2,
y'[0] == 0}, y[0] == w1,
y[x], {x, 0, 30}];
Table[Table[{w1,x,y[x] /. w[w1]}, {w1, 0., 1.0, 0.5}]/. x -> u, {u, 10, 30, 10}]
Output:
{{{0., 10, {3.67814}}, {0.5, 10, {3.25861}}, {1.,10, {2.83907}}},
{{0., 20, {1.18384}}, {0.5, 20, {1.38788}}, {1.,20, {1.59192}}},
{{0., 30, {1.6915}}, {0.5, 30, {1.76862}}, {1.,30, {1.84575}}}}

Answer 2

I see you already chose an answer, but I want to toss this solution for families of linear equations up. Specifically, this is to model a slight variation on Lotka-Volterra.
(*Put everything in a module to scope x and y correctly.*)
Module[{x, y},
(*Build a function to wrap NDSolve, and pass it
the initial conditions and range.*)
soln[iCond_, tRange_, scenario_] :=
NDSolve[{
x'[t] == -scenario[[1]] x[t] + scenario[[2]] x[t]*y[t],
y'[t] == (scenario[[3]] - scenario[[4]]*y[t]) -
scenario[[5]] x[t]*y[t],
x[0] == iCond[[1]],
y[0] == iCond[[2]]
},
{x[t], y[t]},
{t, tRange[[1]], tRange[[2]]}
];
(*Build a plot generator*)
GeneratePlot[{iCond_, tRange_, scen_,
window_}] :=
(*Find a way to catch errors and perturb iCond*)
ParametricPlot[
Evaluate[{x[t], y[t]} /. soln[iCond, tRange, scen]],
{t, tRange[[1]], tRange[[2]]},
PlotRange -> window,
PlotStyle -> Thin, LabelStyle -> Medium
];
(*Call the plot generator with different starting conditions*)
graph[scenario_, tRange_, window_, points_] :=
{plots = {};
istep = (window[[1, 2]] - window[[1, 1]])/(points[[1]]+1);
jstep = (window[[2, 2]] - window[[2, 1]])/(points[[2]]+1);
Do[Do[
AppendTo[plots, {{i, j}, tRange, scenario, window}]
, {j, window[[2, 1]] + jstep, window[[2, 2]] - jstep, jstep}
], {i, window[[1, 1]] + istep, window[[1, 2]] - istep, istep}];
Map[GeneratePlot, plots]
}
]
]
We can then use Animate (or table, but animate is awesome)
tRange = {0, 4};
window = {{0, 8}, {0, 6}};
points = {5, 5}
Animate[Show[graph[{3, 1, 8, 2, 0.5},
{0, t}, window, points]], {t, 0.01, 5},
AnimationRunning -> False]