Let $\lambda$ be a nonzero complex number and let $u(x)$ be some smooth function $\mathbb{R}\to\mathbb{C}$, not identically zero. I want to prove that if $u$ satisfies $$u'' + \lambda u'+ \lambda^2 u = 0 \qquad (*)$$then it doesn't solve some lower-order differential equation, say, $u' + \lambda u = 0$. Now, in this particular example, I can substitute $u' = - \lambda u$ into (*) to get $$ \lambda^2 u - \lambda^2u + \lambda^2u = \lambda^2 u = 0$$ which is false by the hypotheses that neither $u$ nor $\lambda$ is zero. However, this may not work in general, when I have some differential equation satisfied by $u$ and want to prove that $u$ therefore may not satisfy some other differential equation.
(In the case that the equations are linear, I want to say that if $u$ satisfies a given equation of order $n$ that it may not satisfy one of order $m < n$, but I'm not sure yet if I'll be generalizing to a non-linear case as well.)
I therefore would like to learn what I can about the Galois theory, minimality, and factorization of differential equations. I currently have access to Singer and van der Put's Galois Theory of Linear Differential Equations but have so far been unsuccessful in making it through that book, so I'm looking for additional sources.