Parametric plot from two NDsolve solutions

Question

I have two independent ODE systems.

A = NDsolve[..., {x, y}, {t, 0, 10}];
B = NDsolve[..., {a, b}, {t, 0, 10}];

I can draw a ParametricPlot from one ODE. That is,

ParametricPlot[Evaluate[{x[t], y[t]} /. A], {t, 0, 10}]

I wonder if I can draw a ParametricPlot from the two independent ODE systems. That is a ParametricPlot of x[t] taken from A and a[t] taken from B.

Answer 1

Sure you can:

sol1 = NDSolve[{x'[t] == Sin[t], x[0] == 1}, x, {t, 0, 10}];
sol2 = NDSolve[{a'[t] == Cos[t], a[0] == 1}, a, {t, 0, 10}];
ParametricPlot[Evaluate[{x[t], a[t]} /. Flatten@{sol1, sol2}], {t, 0, 10}]

The Flatten is there for the following reason:

{x[t], a[t]} /. {sol1, sol2}
(* {{{InterpolatingFunction[{{0.,10.}},<>][t],a[t]}},   
   {{x[t],InterpolatingFunction[{{0.,10.}},<>][t]}}} *)
{x[t], a[t]} /. Flatten@{sol1,sol2}
(* {InterpolatingFunction[{{0.,10.}},<>][t],
    InterpolatingFunction[{{0.,10.}},<>][t]} *)

You can also do:

ParametricPlot[Evaluate[{x[t] /. First@sol1, a[t] /. First@sol2}], {t, 0, 10}]

Answer 2

Simply you can use the following command:

sol = NDSolve[{x'[t] == Sin[t], a'[t] == Cos[t], a[0] == 1, 
    x[0] == 1}, {x, a}, {t, 0, 10}];

ParametricPlot[{x[t], a[t]} /. sol, {t, 0, 10}, 
 AxesLabel -> {x[t], a[t]}]

Firstly, you may consider one system so as to use the NDSolve only one time and then use the ParametricPlot. It is the same if you add the equations y[t] and b[t], as you say in your question.

So your conclutions are right!