I am a physics student taking ODE this semester.
Let's consider an initial value problem. Consider a homogenous second-order linear differential equation. Suppose we have two solutions $\varphi_1(x)$ and $\varphi_2(x)$. To my understanding, we check the Wronskian to confirm whether we can span the entire solution space of the differential equation with just those two solutions. This is because if the Wronskian is zero, we know that the Wronskian matrix is non-invertible and so is inconsistent or has multiple solutions.
Intuitively, I thought we'd just need to check that $\varphi_1$ and $\varphi_2$ are linearly independent. However, checking the Wronskian seems to impose stricter requirements, i.e., that the vector $(\varphi_1 \ \varphi_2)$ is linearly independent from $(\partial_x\varphi_1 \ \partial_x\varphi_2)$ and equivalently (it seems) that $\partial_x\varphi_1$ is not a multiple of $\partial_x\varphi_2$ and that $\varphi_1$ is not a multiple of $\varphi_2$ (other than trivially when $x_1 = x_2 = 0$). Where the latter statement is what I intuitively think is all that need be checked.
Why do we need to check this stricter Wronskian requirement as opposed to the linear independence of our solutions? Does it have to do with the fact that we need to allow freedom in what the initial value of the first derivative of a solution is? And thus is not about the nature of linear independence, but more about the fact that we're dealing with an initial value problem?