Simple question about Variaton of Parameters in 1 variable

[Remixex]

Given a generic second order ODE, normalized that equals to g(x)
I saw in a video that, when writing the Wronskian, W1 and W2, for writing W1 you had to "delete" the first column if the Determinant and replace it with zeroes until you reach the end, where you replace it with g(x)
Is this always true no matter the initial value problem? (as long as the solution exists of course)

 

[HallsofIvy]

I think you are misunderstanding the purpose. That sounds like it is not directly a matter of "variation of parameters" but specifically a method of solving a system of equations called "Cramer's rule". To find the "i"th unknown in a system of n equations form a fraction with the determinant of coefficients in the denominator and the numerator the same except that the i column is replaced by the right hand side of the system.
For example, if ax+ by = e and cx+ dy= f then
$$x= \frac{\left|\begin{array}{cc} e & b \\ f & d \end{array}\right|}{\left|\begin{array}{cc}a & b \\ c & d \end{array}\right|}$$
and
$$y= \frac{\left|\begin{array}{cc}a & e \\ c & f \end{array}\right|}{\left|\begin{array}{cc}a & b \\ c & d \end{array}\right|}$$