- #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.
[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.