Checking the application of Wronskians in wikipedia for second order ODEs, there is a way to find the other solution $y_2$ if we have found the first solution $y_1$.

The idea is the following:

  1. Suppose that the ODE is of the form $y'' + ay' + by = 0$,
  2. the Wronskian of two functions is $W(x) = y_1y_2' - y_1'y_2$,
  3. using the ODE itself, we can write $W'(x) = -aW(x) \iff W(x) = C_W \exp(-\int a)$,
  4. and finally write $\frac{C_W \exp(-\int a)}{y_1} = y_2' - \frac{y_1'}{y_1}y_2$, which is a first order ODE written in $y_2$. Solve for $y_2$.

The idea seems neat, however I think it's a bit misleading, or perhaps I didn't understand it fully. In the wikipedia page $C_W$ is claimed to be a constant. I don't think that's true at all, $C_W$ depends on $x$, and really it's $C_W(x) = W(x)\exp(\int a)$.

Further, even if we were to write it as $\frac{W(x)}{y_1} = y_2' - \frac{y_1'}{y_1}y_2$, we still can't evaluate $W(x)$ without knowing $y_2$. It's claimed in the wikipedia page that we can use Abel' identity to evaluate the Wronskian without knowing $y_1, y_2$, but it's not clear to me how, since the identity only states that $W(y_1, y_2)(x) = W(y_1, y_2)(x_0) \cdot \exp(-\int_{x_0}^x a)$.



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