# How to decouple a first order ODE system by eliminating coupled functions

Let's see an example of a first order ODE system : \begin{align*} y_1'&=a_{11}\cdot y_1+a_{12}\cdot y_2 \qquad(1)\\ y_2'&=a_{21}\cdot y_1+a_{22}\cdot y_2 \qquad(2) \end{align*} where $y_1$ and $y_2$ are functions we are to solve, and $a_{11},\; a_{12},\; a_{21},\; a_{22}$ are constants. I know 2 methods to solve this, one is to eliminate $y_1$ or $y_2$ by replacing, and then we will get 2 decoupled second order ODEs

$$y_1''-(a_{11}+a_{22})y_1'+(a_{11}a_{22}-a_{12}a_{21})y_1=0$$ the other ODE is similar.

The other method: Try to calculate the eigenvalues and eigenvectors of the coefficient matrix $\{\{a_{11},a_{12}\},\{a_{21},a_{22}\}\}$, and then get the general solutions for system of $y_1$ and $y_2$ simultaneously. Though the second method is more popular in the textbooks, I like the first one more.

So my question is: Which function in Mathematica can decouple the ODE system by the first method automatically?

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In your 2nd equation isn't the first y2 supposed to be a y1 or is it correct as is? –  Sjoerd C. de Vries Nov 4 '12 at 13:30
Changed to y1. Please roll back if I guessed incorrectly. –  Sjoerd C. de Vries Nov 4 '12 at 13:36
It seems to me that going to the eigen-space precisely decouples the equations, so the two methods are equivalent(?) –  chris Nov 4 '12 at 14:35
Do I understand correctly that this question is about manipulating the ODE's for further manual analysis? You don't want to solve them with the help of Mathematica, right? –  sebhofer Nov 4 '12 at 18:06

I believe going to the eigen-space precisely decouples the equations, so the two methods are strictly equivalent.

 mat = {{a11,a12},{a21,a22}};


Now find the eigenvector matrix P (the so called transformation matrix)

 {\[Lambda]s, P} = Eigensystem[mat]; P = Transpose[P];


Let us check that going to the eigen-space makes the Matrix diagonal

 pmat= Inverse[P].mat.P // FullSimplify

(*
1/2 (a11+a22-Sqrt[(a11-a22)^2+4 a12 a21])  0
0  1/2 (a11+a22+Sqrt[(a11-a22)^2+4 a12 a21])
*)


Check that the diagonal elements are the eigenvalues

pmat[[1, 1]]/\[Lambda]s[[1]] // Simplify
pmat[[2, 2]]/\[Lambda]s[[2]] // Simplify


So if we define as new variables {z1[t],z2[t]} so that

Thread[{z1[t],z2[t]}=pmat.{y1[t],y2[t]} ]


The system in z is simply

z1'[t] ==  \[Lambda]s[[1]] z1[t]
z2'[t] ==  \[Lambda]s[[1]] z2[t]


which is decoupled.

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Not really sure why you would want to prescribe the method Mathematica uses to solve the coupled ODEs. If you don't Mathematica solves them just fine:

eq1 = y1'[t] == a11*y1[t] + a12*y2[t];
eq2 = y2'[t] == a21*y1[t] + a22*y2[t];

FullSimplify[DSolve[{eq1, eq2}, {y1[t], y2[t]}, t]]


But if you want to go the eliminate & replace route you could do the following:

eq1D = Assuming[{a11 != 0, a12 != 0, a21 != 0, a22 != 0},
FullSimplify[eq2 /. Solve[eq1, y2[t]] /. D[Solve[eq1, y2[t]], t]]][[1, 1]]


eq2D = Assuming[{a11 != 0, a12 != 0, a21 != 0, a22 != 0},
FullSimplify[eq1 /. Solve[eq2, y1[t]] /. D[Solve[eq2, y1[t]], t]]][[1, 1]]


DSolve[eq1D, y1[t], t]


DSolve[eq2D, y2[t], t]


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