# Discontinuities of coefficient in linear homogeneous system

1. Homework Statement

let $$x^{(1)} = \left( \begin{array}{ccc}1\\1\\1\end{array} \right)$$, $$x^{(2)} = \left( \begin{array}{ccc}1\\t\\t^2\end{array} \right)$$, $$x^{(3)} = \left( \begin{array}{ccc}1\\t\\t^3\end{array} \right)$$

a) Find the Wronskian $$W(x^{(1)}, x^{(2)}, x^{(3)})$$

b) You are told that $$x^{(1)}, x^{(2)}, x^{(3)}$$ are solutions of a

linear homogeneous system $$x' = P(t)x$$

i) what can you say about the discontinuities of the coefficient $$p_{i,j}(t)$$?

ii) Find $$p_{1,1}(t) + p_{2,2}(t) + p_{3,3}(t)$$. (Hint: use your solution to part a).

2. Homework Equations
W = determinant of $$(x^{(1)}, x^{(2)}, x^{(3)})$$

3. The Attempt at a Solution

For part a), the Wronskian is W = determinant of $$(x^{(1)}, x^{(2)}, x^{(3)})$$, which I found to be $$t^4 -2t^3 + t^2$$. I'm not sure how to do part b). What discontinuities is the question refering to, and how do I find them? How do I use my answer to part a) to solve b)ii)? Any help is appreciated!

$t^4- 2t^3+ t^2= t^2(t-1)^2= 0$ when t= 0 or t= 1. Since such an equation has a unique solutions as long as the Wronskian is non-zero, what do you think prevents a solution from being extended past t= 0 or t= 1? It must be discontinuities in matrix P.