We are learning about using the Wronskian matrix to determine linear independence of a set of solutions to find the general solution of an equation:
$$y = c_1 y_1(x) + c_2 y_2(x) +\cdots+c_n y_n(x)$$
where $y_1(x), y_2(x),\ldots,y_n(x)$ are different functions of $x$ that solve the equations and $c_1,c_2,\ldots,c_n$ are arbitrary constants.
I understand how it is possible that there would be multiple solutions to an equation, and how you could use a matrix to make sure all solutions are unique. What I don't understand is how adding your separate solutions automatically results in a final equation that is also a solution, y, or what the arbitrary constants mean in the equation.
In trying to find the answer I looked at the standard form of a linear DVQ:
and I realized I don't know why that is true either. I just took it as a given when it was presented to us on the first day of class.
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