Multiple initial conditions for NDsolve

Question

First, I would like to solve a coupled differential equations. For example:

A0 = {1, 1}; (* initial condition *)
T = 1;  (* time T *)
sol=NDSolve[{a'[t]==a[t]+b[t],b'[t]==a[t]-b[t],{a[0],b[0]}==A0},{a,b},{t,0,T}];
{a[T], b[T]} /. sol[[1]] (* output *)

Then the solution at T is {4.91478, 2.17818}.

Next, I would like to scan the initial condition A0. For example, I would like

A0={Sin[theta],Cos[theta]}

Scan theta from 0 to 2Pi by steps Pi/10, and calculate all the output a[T] and b[T] in a table or saved in other formats. It would be best that the data can be save in this form {{theta_1, a[T]_1,b[T]_1},{theta_2, a[T]_2,b[T]_2}...}. The final result can be plotted, for example, plot "theta - a[T]".

How to achieve this?

Answer 1

Use ParametricNDSolve/ParametricNDSolveValue instead:

pf = ParametricNDSolveValue[
    {a'[t] == {{1, 1}, {1, -1}} . a[t], a[0]=={a0,b0}},
    a,
    {t, 0, 1},
    {a0, b0}
];

(Note, I also converted your equation into vector form). Check that it agrees with your initial A0:

pf[1, 1][1]

{4.91478, 2.17818}

Table output for varying A0:

Table[Flatten[{θ, pf[Sin[θ], Cos[θ]][1]}], {θ, 0, 2 π, π/10}]

{{0, 1.3683, 0.809885}, {π/10, 2.39725, 1.19307}, {π/5, 3.19155, 1.45948}, {(3 π)/10, 3.67343, 1.58302}, {(2 π)/5, 3.79573, 1.5516}, {π/2, 3.54648, 1.3683}, {(3 π)/5, 2.95008, 1.05106}, {( 7 π)/10, 2.0649, 0.630939}, {(4 π)/5, 0.977593, 0.149055}, {( 9 π)/10, -0.205406, -0.347419}, {π, -1.3683, -0.809885}, {( 11 π)/10, -2.39725, -1.19307}, {(6 π)/5, -3.19155, -1.45948}, {( 13 π)/10, -3.67343, -1.58302}, {(7 π)/5, -3.79573, -1.5516}, {( 3 π)/2, -3.54648, -1.3683}, {(8 π)/5, -2.95008, -1.05106}, {( 17 π)/10, -2.0649, -0.630939}, {(9 π)/5, -0.977593, -0.149055}, {( 19 π)/10, 0.205406, 0.347419}, {2 π, 1.3683, 0.809885}}

If you want a plot, you can just use:

Plot[pf[Sin[θ], Cos[θ]][1], {θ, 0, 2 π}]

Answer 2

Table[{theta, {a[T], b[T]} /. NDSolve[{a'[t] == a[t] + b[t], b'[t] == a[t] - b[t], {a[0], b[0]} == {Sin[theta], Cos[theta]}}, {a, b}, {t, 0, T}][[1]]}, {theta, 0., 2 \[Pi], \[Pi]/10}]

That, or, you know, your system can be solved analytically with DSolve.