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I've tried to solve this differential equation system, buy without any luck.

enter image description here

This is my code:

RSolve[{x'[t] == x[t] - y[t], y'[t] == 5 x[t] - 3 y[t], x[0] == 1, y[0] == 2}, {x[t],
y[t]}, t]

Can anyone tell me what I've done wrong?

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You should use DSolve. RSolve is not for differential equations. – m0nhawk Apr 13 '13 at 13:24
Right. RSolve[] is for difference equations; DSolve[] is for differential equations. A matter of continuous versus discrete. – J. M. Apr 13 '13 at 13:52
up vote 5 down vote accepted

I recommend using math: $\ \ \ \dot x=Ax\ \Rightarrow\ x=\text{e}^{At}\ x_0$

 A = {{1, -1}, {5, -3}};
 MatrixExp[A t].{1, 2}   //FullSimplify

gives: $\ \ \ x_1=\text{e}^{-t}\cos{t},\ \ \ x_2=(2+\tan t)\ x_1$

$ $

edit: With one more line, in an intermediate calculation you can can eliminate all six numbers:

 A = {{a, b}, {c, d}};
 MatrixExp[A t].{x1, x2} // FullSimplify

 % /. {a -> 1, b -> -1, c -> 5, d -> -3, x1 -> 1, x2 -> 2} // FullSimplify
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You could also elect to use the two-argument form of MatrixExp[]: MatrixExp[{{a, b}, {c, d}} t, {x1, x2}] – J. M. Apr 13 '13 at 17:21
@Nick Kidman I wonder how to solve this with your method if $x_1(0)=1$ and $x_2(0)=2$ is not given. You can't multiply Matrix A with to vectors of 0. – Jens Jensen Apr 17 '13 at 15:21
@JensJensen: "You can't multiply Matrix A with to vectors of 0." I'm not sure what you're saying, but if you say you can't use it for initial values $(0,0)$, then you're not right. Of course you can: $\text{e}^{At}\vec 0=\vec 0$. And this is exactly the solution of the differential equation: E.g $\dot x_1(0)=x_1(0)-x_2(0)=0$, then it says that the value starts at $x_1(0)=0$ and doesn't change. – NikolajK Apr 17 '13 at 17:54
@NickKidman What if $\dot{x_1} = \dot{x_2}(0)=0$ is given? And you only have the matrix A? – Jens Jensen Apr 17 '13 at 17:57
@JensJensen: I don't know what you're saying. You read off the matrix $A$ from the differential equation. You exponentiate it. You multiply the resulting matrix $\text{e}^{A t}$ with the initial vector. Now matter what the matrix or the differential equations are, this works. If the vector is $\vec 0$, then the vector will stay this $\vec 0$ no matter what the matrix is. If the matrix is four $0'$s, then $\text{e}^{A t}=E$, where $E$ is the unit matrix. No evolution as the differential equation given via $A$ is trivial. For any "What if" question like that, you can also just ask the code here. – NikolajK Apr 17 '13 at 18:00
s = DSolve[{x'@t == x@t - y@t, y'@t == 5 x@t - 3 y@t, x@0 == 1, y@0 == 2}, {x, y}, t]
ParametricPlot[{x[t], y[t]} /. s, {t, 0, 5}, AspectRatio -> 1, PlotRange -> All]

enter image description here

Animate[Graphics[{PointSize[Large], Point[{x[t], y[t]} /. s]}, 
                  PlotRange -> {{-1, 1}, {-2, 2}}, Axes -> True], {t, 0, 10, .01}]

enter image description here

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