Why do we use the function derivatives in the Wronskian?

Question

I'm looking for more information on the matrix that the Wronskian acts on to provide information about a set of solutions to an ODE.

So far I understand that:

• The Wronskian comes from Cramer's rule (at least that is the explanation I see most of the time).
• If the Wronskian is not equal to zero, then the set of solutions is linearly independent.
• If the Wronskian is equal to zero, that the set of solutions could still be linearly independent, but likely isn't.
• If the determinant of a matrix is equal to nonzero, then the matrix is linearly independent, it can be inverted and all that jazz.

What I'm confused on:

• Why did we choose the derivatives of the function in the matrix inside of the Wronskian?
  • I've seen some... lame proofs that the derivatives of differentiable functions should be "linearly independent" to each other. I can easily think of functions that are differentiable that don't follow that rule (like $e^x$).
  • (This might be a stupid question): If the columns of the matrix are not linearly independent then isn't the entire matrix linearly dependent? Is this where our problems of the Wronskian doesn't guarantee linear dependence arise from?

Answer 1

Solving for the coefficients of the solution to (at least) a second order differential equation

$$ y(x)=C_1u(x) + C_2v(x) $$ given the initial conditions: $y(t_0)=y_0$ and $y'(t_0)=y'_0$ means solving for the solution to the matrix below $$ \begin{bmatrix} u(t_0)&v(t_0) \\u'(t_0)&v'(t_0) \end{bmatrix} \begin{bmatrix} C_1\\C_2 \end{bmatrix} = \begin{bmatrix} y_0\\y'_0 \end{bmatrix} $$

which does not have a solution if the Wronskian is zero or the left most matrix has linearly dependent columns or rows (a row or column is a multiple of another row or column respectively).

In the interval $(a,b)$, the complete solution to the higher order linear differential equation must have $u,v$ to be linearly independent on $(a,b)$, the Wronskian not to be zero at some point $t_0$ in $(a,b)$ or never zero at all. If one of these conditions are met, the solution is called a fundamental solution set on the ODE at $(a,b)$.