Wronskian and Homogenous Equation and Absolute Value

Question

I was given these two functions: $f_1(x)=2+x \ \text{and} \ f_2(x)=2+\lvert{x}\rvert$ An interval as well $I_0=(-\infty,\infty)$

I began to setup my Wronskian as follows:$$W=\begin{bmatrix}2+x&2+\lvert{x}\rvert \\ 1&\frac{x}{\lvert{x}\rvert}\end{bmatrix}$$

Then I began to calculate the determinant of W: $$\det(W)=\frac{2x+x^2}{\lvert{x}\rvert}-2-\lvert{x}\rvert$$

My Question

My question is this the $\det(W)$ is equal to zero, I graphed it and got this I see that in its domain that it is linearly independent for a subset of that domain which is $I=(-\infty,0)$, and for the other subset it is linearly dependent that subset being of course $I_1=(0, \infty)$ Would my analysis be right or would one plainly say that it is linearly dependent on the whole interval $I_0$.

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Answer 1

The Wronskian is undefined for $x=0.$ The rows/columns of the Wronskian are linearly independent on $(- \infty,0)$ and linearly dependent on $(0,\infty).$