# Wronskians in ODE

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 $$W(y_1, y_2)(x) = W(y_1, y_2)(x_0) \cdot \exp(-\int_{x_0}^x a)$$.