Find the interval on which the function are linearly dependent and linearly independent 
When I find the wronskians of this function then I get linearly dependent in the interval but in my homework solutions it's given that is linearly dependent on the interval but linearly independent on R but I don't understand how it could possible to be linearly independent please help
 A: Go back to the definition. For two elements of a vector space, such as $f$ and $g$ in this example, being linearly dependent means that there exist scalars $c_1$ and $c_2$, not both zero, such that
$$c_1f+c_2g=0.$$
Note that since we're dealing with a vector space of functions here, the "$0$" on the right-hand side means the identically zero function. In other words, the equation
$$c_1f(x)+c_2g(x)=0$$
must be true for all $x\in\mathbb{R}$ with the same $c_1,c_2$ (not both zero). But with the given functions this is impossible, so they are linearly independent by definition.
And why is it impossible? Roughly speaking, because you would have to use different values of $c_1$ and $c_2$ for positive $x$ and for negative $x$. In other words: in this example, the two functions are linearly dependent on $(-\infty,0)$ and on $[0,+\infty)$ separately. But those dependencies are different from each other (with different coefficients), so there's no common dependence that would hold for all $x\in(-\infty,+\infty)$.
