suppose I have some equation: $$ \frac{dy^2}{dx^2} + (x^2-3x)y' + 12y = 0 $$
(The above equation is just for example)
Using Abel's Theorem for the Wronskian: $$ W(y_1, y_2)(x) = ce^{-\int p(x)~dx} $$
If I know the value of the Wronskian for some $x = a$, can I calculate the Wronskian for any value of $x$?
For instance, suppose I know for $x = 5$, $W(y_1, y_2) (5) = 27$ (I know this is not the case, but let's suppose), where $y_1$ and $y_2$ are some solution to the equation.
Consequently, can I solve for the constant C:
$27 = ce^{-\int x^2 - 3x~dx} , x = 5$
$27 = ce^{-\dfrac{x^3}{3} + \dfrac{3x^2}{2}} , x = 5$
$27 = ce^{-\dfrac{x^3}{3} + \dfrac{3x^2}{2}} , x = 5$
$27 = ce^{-\dfrac{25}{3}} $
$ c = 27e^{\dfrac{25}{3}} $
And hence, would I be able to solve for all W(y1, y2)(x)?