Fundamental matrix of a second order 2x2 system of ODEs

In summary, the equation for the fundamental matrix for a 4x4 first order system can be written as\begin{pmatrix}\mathbf{x}\\ \mathbf{x}' \end{pmatrix}' = \begin{pmatrix} 0 & I \\ A & 0 \end{pmatrix}where the eigenvector corresponding to the eigenvalue of mu with eigenvalue \lambda is \mathbf{v} with eigenvalue \mu^2.
  • #1
EinsteinCross
1
0
Let ## \mathbf{x''} = A\mathbf{x} ## be a homogenous second order system of linear differential equations where

##
A = \begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
## and ##
\mathbf{x} = \begin{bmatrix}
x(t)\\ y(t))
\end{bmatrix}
##

Now to solve this equation we transform it into a 4x4 first order system ##\mathbf{X'} = M\mathbf{X} ## where

##
M = \begin{bmatrix}
0 & 1 & 0 & 0 \\
a & 0 & b & 0\\
0 & 0 & 0 & 1 \\
c & 0 & d & 0
\end{bmatrix}## and ##
\mathbf{X} = \begin{bmatrix}
x(t)\\ x'(t)
\\ y(t)
\\ y'(t)
\end{bmatrix}
##

Now solving the first order system by calculating the eigenvalues and corresponding eigenvectors gives us the fundamental matrix for the 4x4 system
##
\Phi (t) = [[\xi_{1}],[\xi_{2}],[\xi_{3}],[\xi_{4}]]
\begin{bmatrix}
e^{\lambda_{1}t }
\\ e^{\lambda_{2}t }
\\ e^{\lambda_{3}t }
\\ e^{\lambda_{4}t }
\end{bmatrix}^{T} ## where the xi brackets are the eigenvectors in column form.

But my question is this: What I am looking to do is to find the fundamental matrix of the original second order 2x2 system which is represented as ##
\Psi (t) = \begin{bmatrix}
\psi_{1}(t) & \psi_{2}(t)
\\ \psi_{3}(t)
& \psi_{4}(t)
\end{bmatrix} ## such that ##\Psi ''(t) = A\Psi (t)##. So how does one extract the solution(s) to the original 2x2 system from the fundamental matrix of the first order 4x4 system?
 
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  • #2
It is better to construct your first order system in block matrix form as [tex]
\begin{pmatrix} \mathbf{x} \\ \mathbf{x}' \end{pmatrix}' = \begin{pmatrix} 0 & I \\ A & 0 \end{pmatrix}
\begin{pmatrix} \mathbf{x} \\ \mathbf{x}' \end{pmatrix}.[/tex] Now consider an eigenvector [itex](\mathbf{v}\, \mathbf{u})^T[/itex] of this with eigenvalue [itex]\mu[/itex]. By definition [tex]
\mu \begin{pmatrix} \mathbf{v} \\ \mathbf{u} \end{pmatrix} = \begin{pmatrix} 0 & I \\ A & 0 \end{pmatrix}
\begin{pmatrix} \mathbf{v} \\ \mathbf{u} \end{pmatrix}[/tex] so that [tex]\left. \begin{array}{c}
\mu\mathbf{v} = \mathbf{u} \\
\mu\mathbf{u} = A\mathbf{v} \end{array}\right\}\quad\Rightarrow\quad A\mathbf{v} = \mu^2 \mathbf{v}.[/tex] Hence [itex]\mu^2 = \lambda[/itex] is an eigenvalue of [itex]A[/itex] with [itex]\mathbf{v}[/itex] its corresponding eigenvector. This leads to a solution of the form [tex]
\mathbf{x}(t) = \mathbf{v}(Ae^{\sqrt{\lambda}t} + Be^{-\sqrt{\lambda}t})[/tex]
 
Last edited:
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  • #3
An equivalent approach is to leave it as a system of second-order ODEs. Start with
$$ \mathbf{x''} = A\mathbf{x} $$
and assume a solution of the form ##\mathbf{x} = \mathbf{x_0} \, e^{\lambda t}##. You then get
$$ \lambda^2 \mathbf{x_0} = A\mathbf{x_0}. $$
which is a standard eigenvalue problem. This is the approach I always use for equations of this form.

jason
 
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1. What is the fundamental matrix of a second order 2x2 system of ODEs?

The fundamental matrix of a second order 2x2 system of ODEs is a matrix that contains the solutions to the system of equations. It is used to find the general solution to the system of ODEs.

2. How is the fundamental matrix of a second order 2x2 system of ODEs calculated?

The fundamental matrix is calculated by finding the eigenvalues and eigenvectors of the coefficient matrix of the system of ODEs. These values are then used to construct the fundamental matrix.

3. What is the importance of the fundamental matrix in solving ODEs?

The fundamental matrix is important because it provides a way to find the general solution to a system of ODEs. It also allows for the calculation of the solution at any point in time.

4. Can the fundamental matrix of a second order 2x2 system of ODEs be used for higher order systems?

Yes, the fundamental matrix can be extended to higher order systems of ODEs. However, the size of the matrix will increase as the order of the system increases.

5. Are there any limitations to using the fundamental matrix to solve ODEs?

One limitation is that the fundamental matrix can only be used for linear systems of ODEs. It also assumes that the initial conditions are known and that the system is time-invariant.

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