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*](http://rads.stackoverflow.com/amzn/click/3540442286) but have so far been unsuccessful in making it through that book, so I'm looking for additional sources.



What are some good references on the Galois theory, factorization, or minimality of differential equations?
You mentioned already the book of van der Put and Singer, which is a standard reference for the Galois Theory of linear differential equations.


Other books on the topic include:


* Teresa Crespo and Zbigniew Hajto. Algebraic groups and differential Galois theory, volume 122 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2011.
* Irving Kaplansky. An Introduction to Differential Algebra. Hermann, Paris, 2nd edition edition, 1976.
* Andy R. Magid. Lectures on differential Galois theory, volume 7 of University Lecture Series. American Mathematical Society, Providence, RI, 1994.


