# Wronskian to determine lin.ind. of solutions to a system of ODEs

1. Mar 23, 2013

### SithsNGiggles

1. The problem statement, all variables and given/known data

In my book, I'm given that $\vec{x}_1=\left(\begin{matrix}t^2\\t\end{matrix}\right), \vec{x}_2=\left(\begin{matrix}0\\1+t\end{matrix}\right), \vec{x}_3=\left(\begin{matrix}-t^2\\1\end{matrix}\right)$ are solutions. My textbook presents an algebraic way to show that the vectors are linear independent, but I was hoping to see if I can use the Wronskian to show the same result.

2. Relevant equations

3. The attempt at a solution

I thought this was how the Wronskian would look:
$W\left(\vec{x}_1, \vec{x}_2, \vec{x}_3\right)= \left|\begin{matrix} \left(\begin{matrix}t^2\\1\end{matrix}\right) & \left(\begin{matrix}0\\1+t\end{matrix}\right) & \left(\begin{matrix}-t^2\\1\end{matrix}\right)\\ \left(\begin{matrix}2t\\0\end{matrix}\right) & \left(\begin{matrix}0\\1\end{matrix}\right) & \left(\begin{matrix}-2t\\0\end{matrix}\right)\\ \left(\begin{matrix}2\\0\end{matrix}\right) & \left(\begin{matrix}0\\0\end{matrix}\right) & \left(\begin{matrix}-2\\0\end{matrix}\right) \end{matrix}\right|$

But I couldn't see how to proceed from there, since matrix multiplication won't work. How would I find this determinant?

2. Mar 24, 2013

### vela

Staff Emeritus
You can't because it doesn't make sense.

In any case, you should be able to see by inspection that the three vectors aren't linearly independent.

3. Mar 24, 2013

### SithsNGiggles

Okay, thanks.

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