Hello Vedat, I am afraid I do not understand what you mean. What do you mean by "one gets the value of x[t] versus G" ? x[t] is a function and not a value! x[ .85 ] for example is a value, and this value certainely depends on G. But for a given t x[ t ] is a linear function of G (look for example at fx[.1, g] ), so you get a straight line. The same is true for y,
I attach some code which shows how the functions x[t] and y[t] ( and not a value ) change with G (in my code g)
eq1 = x'[t] == y[t];
eq2 = y'[t] == 6 x[t] - y[t] + g;
sol = DSolve[{eq1, eq2, x[0] == 0, y[0] == 1}, {x, y}, t];
fx[t_, g_] := Evaluate[(x /. sol[[1, 1]])[[2]]]
fy[t_, g_] := Evaluate[(y /. sol[[1, 2]])[[2]]]
(* Show x[t] depending on g *)
Manipulate[
Plot[fx[tt, g], {tt, 0, 2}, PlotRange -> {0, 6}],
{g, 0, 1}]
(* Show both x[t] (red) y[t] (blue) depending on g *)
Manipulate[
Plot[{fx[t, g], fy[t, g]}, {t, 0, 2},
PlotRange -> {0, 6},
PlotStyle -> {Red, Blue}],
{g, 0, 1}]
(* Show the vector { x[t], y[t] } depending on g *)
Manipulate[
ParametricPlot[{fx[t, g], fy[t, g]}, {t, 0, 2},
PlotRange -> {{0, 2}, {0, 4}}],
{g, 0, 1}]