# Plotting a function that uses the solutions to a system of differential equations

I am trying to draw a plot by using the solution of differential equations

z = 0;
Ω = 2.2758;
τ = 13.8;
T2 = 200;
ω0 = 1;
r = 0.7071;
Δ = 1.7758;
ΩR = 2.2758;
ω == 0.5;
system = {u'[t] ==  Ω*v[t],
v'[t] ==  Ω*u[t] - 2*ΩR*w[t] - v[t]/T2,
w'[t] ==  2*ΩR*v[t]};
initialvalues = {u[0] ==  0, v[0] ==  0, w[0] ==  -1};
sol = DSolve[Join[system, initialvalues], {u[t], v[t], w[t]}, t]
FT = (-2*2.2758*r*u[t])/ω0^2 E^(-(r^2/ω0^2) - (t^2/τ^2))* Cos[kz - ωt];
p1 =
Plot[FT, {t, -20, 20},
FrameLabel -> {"t", "FT"},
Frame -> True,
RotateLabel -> True,
PlotRange -> {-0.1, 0},
PlotStyle -> {Thickness[0.002]}]

right now I went to plot "FT" by using the result of "u" , How I can do that ?

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Sorry for the mess earlier, I hope the question is clear now – saysics Oct 5 '13 at 12:32
There are several syntax errors and mistakes in the code, and as there is nothing much else to this question than that, I'm voting to close. A remaining problem is that kz is not defined. – Oleksandr R. Oct 15 '13 at 17:55
Your solution from DSolve is complex. How do you want to plot it in the real plane? You can try to plot Plot[Re[FT /. sol /. {kz -> .1, \[Omega]t -> .2}], {t, -20, 20}] to get an idea, but I only guessed at the kz and \[Omega]t values. – István Zachar Oct 16 '13 at 7:54

Here is one way to make your plot with results from solving your system of ODEs.

z = 0;
Ω = 2.2758;
τ = 13.8;
T2 = 200;
ω0 = 1;
r = 0.7071;
Δ = 1.7758;
ΩR = 2.2758;

ω = 0.5;

Note: syntax error corrected in expresion for ω.

system = {
u'[t] == Ω*v[t],
v'[t] == Ω*u[t] - 2*ΩR*w[t] - v[t]/T2,
w'[t] == 2*ΩR*v[t]
};
initialvalues = {u[0] == 0, v[0] == 0, w[0] == -1};
sol = DSolve[Join[system, initialvalues], {u[t], v[t], w[t]}, t] // Chop // Simplify;

I used Chop and Simplify to get rid of some fuzz produced by DSolve because it was given approximate numbers.

FT = (-2*2.2758*r*u[t])/ω0^2 E^(-(r^2/ω0^2) - (t^2/τ^2))*Cos[k*z - ω*t]
-3.21844 E^(-0.49999 - 0.005251 t^2) Cos[0.5 t] u[t]

Note: syntax errors corrected in the expression for FT.

Plot[FT /. sol, {t, -20, 20}]

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