Here is the theorem I use:
Two solutions $\phi_1$, $\phi_2$ of $L(y)=y''+a_1y'+a_2y=0$, where $a_1$ and $a_2$ are constants, are linearly independent on an interval $I$ if, and only if, the Wronskain $W(\phi_1,\phi_2)\ne0$ for all $x\in I.$
Then, consider the functions: $$\phi_1(x)=x,\phi_2(x)=|x|, x\in(-\infty,\infty). $$Are they linearly dependent ot independent?
My answer is that they are linearly dependent since when $x\ge0$, $\phi_1(x)=x=\phi_2(x)$ and I plug them and the corresponding derivatives into the Wronskian, $W=0$ for all $x\ge0$; and also check when $x\lt0$, the Wronskian also equals to 0, thus $\phi_1$ and $\phi_2$ here are linear dependent. But the answer in the back of the book says they are linearly independent. Where went wrong?
A similar example is $$\phi_1(x)=x^2, \phi_2(x)=x|x|, x\in(-\infty,\infty).$$The answer is also linear independence, but I think they are linear dependence.
Thanks for your answer.