Solution to Matrix Differential Equation

In summary, the conversation discusses how to write the form of the solution to a given equation involving a system of differential equations. The solution involves finding a matrix D that is similar to the original matrix A, and using linear algebra concepts such as eigenvectors and eigenvalues to solve the system. The conversation also mentions the need for a good understanding of linear algebra in order to solve such systems of differential equations.
  • #1
WolfOfTheSteps
138
0
How do I write the form of the solution to this equation:

[tex]

\dot{\vec{x}}(t) =
\left [ \begin{array}{cc}
a_{11}(t) & a_{12}(t) \\
a_{21}(t) & a_{22}(t)
\end{array} \right ] \vec{x}(t)

[/tex]


I just need to be able to write x1(t) and x2(t) so I can do the rest of the problem I'm working on. Getting this would just be a small step in my solution, but I am very rusty with my differential equations! :(

Initially, I thought to write:

[tex]
x_1(t) = \int_{t_0}^{t}x_1(\tau)a_{11}(\tau) + x_2(\tau)a_{12}(\tau)d\tau
[/tex]


[tex]
x_2(t) = \int_{t_0}^{t}x_1(\tau)a_{21}(\tau) + x_2(\tau)a_{22}(\tau)d\tau
[/tex]

But that has the solutions with dependence on x1(t) and x2(t). That's not the way to write it, is it?

Thanks.
 
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  • #2
WolfOfTheSteps said:
How do I write the form of the solution to this equation:

[tex]

\dot{\vec{x}}(t) =
\left [ \begin{array}{cc}
a_{11}(t) & a_{12}(t) \\
a_{21}(t) & a_{22}(t)
\end{array} \right ] \vec{x}(t)

[/tex]


I just need to be able to write x1(t) and x2(t) so I can do the rest of the problem I'm working on. Getting this would just be a small step in my solution, but I am very rusty with my differential equations! :(

Initially, I thought to write:

[tex]
x_1(t) = \int_{t_0}^{t}x_1(\tau)a_{11}(\tau) + x_2(\tau)a_{12}(\tau)d\tau
[/tex]


[tex]
x_2(t) = \int_{t_0}^{t}x_1(\tau)a_{21}(\tau) + x_2(\tau)a_{22}(\tau)d\tau
[/tex]

But that has the solutions with dependence on x1(t) and x2(t). That's not the way to write it, is it?

Thanks.

Let's look at your system of DEs this way:
[tex]
\dot{\vec{x}}(t) = A \vec{x}(t)
[/tex]

What you'd like is a system that looks like this:
[tex]
\dot{\vec{x}}(t) = D \vec{x}(t)
[/tex]
where D is a diagonal matrix.

This will untangle things so that you have x1'(t) = d11 x1(t) and x2'(t) = d22 x2(t).

These are easy to solve, since each one involves only a single variable.

Getting the matrix D is the hard part, though, since doing this involves changing to a different basis (for R2 in your case). Without going into too many details, you'll want to find a matrix D that is similar to your original matrix A, which I'm assuming is invertible. Similarity is precisely defined this way: If A ~ D, then for some invertible matrix P, AP = PD.

Equivalently, P[tex]^{-1}[/tex]AP = P[tex]^{-1}[/tex]PD = D.

You will need to come up with a matrix P whose columns are the new basis, and a matrix P[tex]^{-1}[/tex], the inverse of P.

To wind this up, the columns of P are the eigenvectors of matrix A, and it turns out that the diagonal entries of D are the eigenvalues of A.

I hope I've given you enough to at least get you started searching for the things to learn more about. Solving a system of DEs, even the simplest possible system in two variables requires a significant amount of understanding in linear algebra.

Mark
 
  • #3
Mark44 said:
Let's look at your system of DEs this way:
[tex]
\dot{\vec{x}}(t) = A \vec{x}(t)
[/tex]

What you'd like is a system that looks like this:
[tex]
\dot{\vec{x}}(t) = D \vec{x}(t)
[/tex]
where D is a diagonal matrix.

This will untangle things so that you have x1'(t) = d11 x1(t) and x2'(t) = d22 x2(t).

These are easy to solve, since each one involves only a single variable.

Getting the matrix D is the hard part, though, since doing this involves changing to a different basis (for R2 in your case). Without going into too many details, you'll want to find a matrix D that is similar to your original matrix A, which I'm assuming is invertible. Similarity is precisely defined this way: If A ~ D, then for some invertible matrix P, AP = PD.

Equivalently, P[tex]^{-1}[/tex]AP = P[tex]^{-1}[/tex]PD = D.

You will need to come up with a matrix P whose columns are the new basis, and a matrix P[tex]^{-1}[/tex], the inverse of P.

To wind this up, the columns of P are the eigenvectors of matrix A, and it turns out that the diagonal entries of D are the eigenvalues of A.

I hope I've given you enough to at least get you started searching for the things to learn more about. Solving a system of DEs, even the simplest possible system in two variables requires a significant amount of understanding in linear algebra.

Mark

Yes, my linear algebra is a few years out of service, but it is not non-existent. You have refreshed my memory. Thanks.
 

1. What is a matrix differential equation?

A matrix differential equation is an equation that involves a matrix-valued function and its derivatives. It is a generalization of a regular differential equation, where the unknown function is a scalar.

2. What are the applications of matrix differential equations?

Matrix differential equations have various applications in physics, engineering, and economics. They are used to model dynamic systems such as population growth, chemical reactions, and electronic circuits.

3. How do you solve a matrix differential equation?

The solution to a matrix differential equation can be found by finding the eigenvalues and eigenvectors of the coefficient matrix, and then using the matrix exponential function. Alternatively, numerical methods such as the Runge-Kutta method can also be used to solve matrix differential equations.

4. Can matrix differential equations have multiple solutions?

Yes, matrix differential equations can have multiple solutions. This is because the solution is dependent on the initial conditions, and different initial conditions can lead to different solutions.

5. Are there any special types of matrix differential equations?

Yes, there are several special types of matrix differential equations, such as linear, nonlinear, and autonomous matrix differential equations. These types have different properties and require different solution methods.

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