Wronskian and Homogenous Equation and Absolute Value

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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1 Answer

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