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Solving a system of ODEs

116
2
Given the matrix [itex]b=\begin{pmatrix}-1&0&-1\\-4&3&-1\\0&0&-2\end{pmatrix}[/itex] decide if the system of ODEs, [itex]\frac{dx}{dt}=Bx[/itex] is decoupled. If yes find the general solution x=xh(t)

2. Homework Equations



3. The Attempt at a Solution
I would say the matrix is decoupled since the second equation involving [itex]2x[/itex]2(t) can be solved without the other two equations. Then the third equation can be solved without knowing [itex]x[/itex]1(t). We have:
[itex]
x'_1 = -x_1 - x_3 \\
x'_2 = -4x_1 + 3x_2 - x_3 \\
x'_3 = -2x_3
[/itex]
Im not sure where to go from here.
 

Filip Larsen

Gold Member
1,216
158
You may want to look into matrix diagonalization [1]. If B can written as A D A-1, where D is a diagonal matrix, can you then use this to rewrite you ODE system to a new uncoupled variable basis?

[1] http://en.wikipedia.org/wiki/Diagonalizable_matrix
 
116
2
I have found the diagonal matrix,
[tex]D=
\begin{pmatrix}
-1 & 0 & 0\\
0 & 3 & 0\\
0 & 0 & -2
\end{pmatrix}
[/tex]
I thought the matrix B was already uncoupled though. Is this not the case?
 
32,631
4,377
Given the matrix [itex]b=\begin{pmatrix}-1&0&-1\\-4&3&-1\\0&0&-2\end{pmatrix}[/itex] decide if the system of ODEs, [itex]\frac{dx}{dt}=Bx[/itex] is decoupled. If yes find the general solution x=xh(t)

2. Homework Equations



3. The Attempt at a Solution
I would say the matrix is decoupled since the second equation involving [itex]2x[/itex]2(t) can be solved without the other two equations.
???
Do you mean the third equation? It involves only x3' and x3.
Then the third equation can be solved without knowing [itex]x[/itex]1(t). We have:
[itex]
x'_1 = -x_1 - x_3 \\
x'_2 = -4x_1 + 3x_2 - x_3 \\
x'_3 = -2x_3
[/itex]
Im not sure where to go from here.
I have found the diagonal matrix,
[tex]D=
\begin{pmatrix}
-1 & 0 & 0\\
0 & 3 & 0\\
0 & 0 & -2
\end{pmatrix}
[/tex]
I thought the matrix B was already uncoupled though. Is this not the case?
The system of equations was not uncoupled. The purpose of finding a diagonal matrix that is similar to B gives you a system that is uncoupled. In an uncoupled system, each equation involves only a single variable and its derivative.
 
116
2
That was an error on my part, what I meant was the matrix is decoupled since the second equation involving [itex]-2x_3(t)[/itex] can be solved without the other two equations and then we can solve for [itex]x_1(t)[/itex] without knowing [itex] x_2(t)[/itex]
 

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