Math Help - Interval

1. Interval

Determine the largest interval where there is guaranteed to be a unique solution for the following IVP. Justify why this is the largest interval.

$\textbf{X}'=\left(\begin{array}{cc} \ln t & 6 \\ 3t^2 & \sin t\end{array}\right)\textbf{X} + \left(\begin{array}{c} \sqrt{5-t}\\ \cot t \end{array}\right), \textbf{X}(\frac{5\pi}{2})=\left(\begin{array}{c}0 \\1 \end{array}\right)$

2. Can anyone tell me the best/easiest way to go about finding this interval?

3. Originally Posted by caeder012
Determine the largest interval where there is guaranteed to be a unique solution for the following IVP. Justify why this is the largest interval.

$\textbf{X}'=\left(\begin{array}{cc} \ln t & 6 \\ 3t^2 & \sin t\end{array}\right)\textbf{X} + \left(\begin{array}{c} \sqrt{5-t}\\ \cot t \end{array}\right), \textbf{X}(\frac{5\pi}{2})=\left(\begin{array}{c}0 \\1 \end{array}\right)$
The interval you seek is the largest interval containing the initial point
on which the derivative is finite and single valued (or assume are allowed to
make a choice about which branch of $\sqrt{}$ you are on).

RonL

4. Originally Posted by CaptainBlack
The interval you seek is the largest interval containing the initial point
on which the derivative is finite and single valued (or assume are allowed to
make a choice about which branch of $\sqrt{}$ you are on).

RonL
Alright, well I've put some thought into this problem. I also asked my prof about it. What I found interesting in her reply was that she said I should look at theorems and that I DONT have to solve a system to know whether a unique sol'n exists. Looking at your response, we're given the derivative. That is, X' = ... .

Looking at theorems in my book, this may be helpful:

(Note that the form I gave was X' = AX + F(t)).

So, the theorem says to let the entries of the matrices A(t) and F(t) be functions that are continuous on a common interval I that contains the point t_0. Then there exists a unique sol'n to the IVP X' = A(t)X + F(t) subject to X(t_0) = X_0 on the interval.