ParametricPlot[{{2  Cos[t], Sin[t]}, {Cos[t], Sin[t]}}, {t, 0, 2  Pi}]

\(\(
A = {{1, 2}, {3, 4}}\)\)

% // MatrixForm

Eigensystem[%]

<< Graphics`PlotField`

\(\(
PlotVectorField[{x1 + 2  x2, 3  x1 + 4  x2}, {x1, \(-10\), 10}, {
        x2, \(-10\), 10}]\)\)

\(\(
Clear[x1, x2]\)\)

\(DSolve[{\(x1'\)[t] == x1[t] + 2  x2[t], 
	
        \(x2'\)[t] == 3  x1[t] + 4  x2[t]}, \ {x1[t], x2[t]}, t]\)

y1[t_] = x1[t] /. % // First; y2[t_] = x2[t] /. % // First;

sol = {y1[t], 
          y2[t]} /. {{C[1] -> 0, C[2] -> 1}, {C[1] -> 1, C[2] -> 0}, 
           {C[1] -> 1, C[2] -> 1}, {C[1] -> -1, C[2] -> -1}}

ParametricPlot[Evaluate[sol], {t, 0,  .2}]

\(sub = {}; 
    For[i = 0, i < 10, \(i++\), 
	
      For[j = 0, j < 10, \(j++\), 
		
        sub = Append[
            sub, {C[1] -> 2 \((i/10 - 1/2)\), C[2] -> j/10 - 1/2}]]]; \)

sub

sol = {y1[t], y2[t]} /. sub;

ParametricPlot[Evaluate[sol], {t, - .5,  .5}]