The method of undetermined coefficients requires, that the input of the ODE be a function that returns to some linear combination of itself if you take the derivative enough times.
Tangent is offered as an example of a function that fails this condition. The derivatives of the tangent produce functions containing higher and higher powers of $\tan^n x$. My books says they are independent, or rather, that new independent terms will always be produced by taking the derivative.
I tried to check this idea with the wronskian of $\tan^n x$ and $\tan^m x$, but I got a function that is often zero,
so it is inconclusive. How can I simply prove independence?
Also, why is the tangent so badly behaved compared to sin and cosine? I don't know if there is a succinct answer to that, but I thought I'd ask anyway.