Consider the equation $-cS' + SS' + \delta S''' - \epsilon S'' = 0$. The function $S = S(\xi, \delta, \epsilon)$, where $\xi = x - ct$ satisfies this ODE. In this case, ' denotes differentiation with respect to $\xi = x - ct$.
I am trying to verify this result by performing the differentiation myself but I don't really understand how to differentiate S. Should I apply the chain rule or the product rule?
This result is from a research paper I am currently reading and attempting to understand.
Comment too long to be edited in the comments section :
$$-cS' + S\,S' + \delta \,S''' - \epsilon\,S'' =0\tag 1$$ Since you wrote "$\:\:'$ denotes differentiation with respect to $\xi\:$ " the variable is $\xi\text{ and } c,\delta,\epsilon$ are parameters. So the ODE might be : $$-c\frac{dS}{d\xi}+S(\xi)\frac{dS}{d\xi}+\delta\frac{d^3S}{d\xi^3}-\epsilon\frac{d^2S}{d\xi^2}=0$$ This ODE is not linear. Solving it analytically would not be elementary but requiring high level maths.
Probably there is a typo in Eq.$(1)$
For maths at student level the equation might be : $$-c\frac{dS}{d\xi}+S(\xi)+\frac{dS}{d\xi}+\delta\frac{d^3S}{d\xi^3}-\epsilon\frac{d^2S}{d\xi^2}=0$$ with some coefficients for $S(\xi)$ and/or for $\frac{dS}{d\xi}$ that one cannot guess.
Check the wording of the problem and the equation.