General Solution of x'(t)=Ax(t)

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July 26th 2009, 01:13 PM
diddledabble

General Solution of x'(t)=Ax(t)

General Solution of x'(t)=Ax(t)

$<br /> A=\begin{array}{cc}1&3\\12&1\end{array}<br />$
July 26th 2009, 02:13 PM
pickslides

You need to solve these DEs

$x_1'(t) = x_1(t)+3x_2(t)$

$x_2'(t) = 12x_1(t)+x_2(t)$
July 26th 2009, 11:52 PM
Moo

Hello,

Treat A as a constant (which it is).

We know that the general solution of this type of equation is $x(t)=e^{At}$

And when A is a matrix, $e^{At}=\sum_{n\geq 0} \frac{A^n t^n}{n!}$

See here : http://www.mathhelpforum.com/math-he...al-matrix.html for an example of computing this exponential.

If you don't understand, I can explain.
July 27th 2009, 01:13 AM
CaptainBlack

Quote:

Originally Posted by diddledabble

General Solution of x'(t)=Ax(t)

$<br /> A=\begin{array}{cc}1&3\\12&1\end{array}<br />$

You can also convert this into a pair of uncoupled second order homogeneous constant coefficient ODE's.

To do this write them out in full as two first order ODE's differnetiate one of them again and substitute the derivative from the other to eliminate one of the variables ...

CB
July 29th 2009, 02:53 PM
diddledabble

Still don't get it

I still am not sure what to do with the matrix.
July 31st 2009, 04:56 PM
Haven

EigenValue Method

Okay here's an easy way to solve it:

$x'(t) = Ax(t)$

We assume the form of the solution that $x(t) = v(t)*e^{rt}$

It turns out the r, corresponds to the eigenvalue of the matrix A and $v(t)$ are the eigenvectors

$A=\left[\begin{array}{cc}1&3\\12&1\end{array}\right]$

$det(A - rI) = \left|\begin{array}{cc}1-r&3\\12&1-r\end{array}\right| = r^2 -2r -35 = (r+5)(r-7) = 0$ for an aribitrary constant $c_1$

so the roots of this polynomial are: $r = -5, 7$

Now we plug our values of r back in
$A - (-5)I = \left[\begin{array}{cc}6&3\\12&6\end{array}\right]\\\rightarrow\\\left[\begin{array}{cc}2&1\\0&0\end{array}\right]$

So we see that $-2v_1 = v_2$ , which means our eigenvector is $v_1(t) = \left[\begin{array}{cc}1\\-2\end{array}\right]$ and our first solution is $x_1(t) = \left[\begin{array}{cc}1\\-2\end{array}\right]\\e^{-5t}$

Similarly for $r =7$
we compute $A - (7)I = \left[\begin{array}{cc}-6&3\\12&-6\end{array}\right]\\\rightarrow\\\left[\begin{array}{cc}-2&1\\0&0\end{array}\right]$

eigenvector: $v_1(t) = \left[\begin{array}{cc}1\\2\end{array}\right]$

second solution $x_2(t) = \left[\begin{array}{cc}1\\2\end{array}\right]\\e^{7t}$

Our final solution is a linear combination to the two solutions $x(t) = {c_1}x_1(t) + {c_2}x_2(t) = c_1\left[\begin{array}{cc}1\\-2\end{array}\right]\\e^{-5t}\\+\\c_2\left[\begin{array}{cc}1\\2\end{array}\right]\\e^{7t}$

Where $c_1, c_2$ are arbitrary constants