While studying Maxwell's derivation for the distribution of molecule speeds in an ideal gas, I asked myself the following question: What would be the solution to a differential equation of the form: $$f(x) - F(u(x)) = 0, $$ where $F' = f, $ $u(x) = x^n$ and $n \in \mathbb{N}$.
Right off the bat I can see that $F(x) = e^x$ for $x=0$ and $x=1$ is a solution (I'm not quite sure though).
I must admit I am a mathematician hobbyist, and my formal education is in Computer Science (inlcuding the maths needed for the degree + 3 levels of calculus). However, I have been a student of mathematics for almost a full decade now. I have experience with ordinary differential equations and limited experience with partial differential equations. The former being revisited and reviewed in order to properly understand what the equations of physics communicate. Now, I have not found much on differential equations involving this sort of functional composition.
(PS: This is not a physics question)
I tried this case: $$ F'(x) = F(x^2), F(0)=1. $$ There is a series solution like this \begin{align} F(x) &= 1+x+{\frac{1}{3}}{x}^{3}+{\frac{1}{21}}{x}^{7}+{\frac{1}{315}}{x}^{15 }+{\frac{1}{9765}}{x}^{31}+\dots \\ &= 1 + \frac{x}{1}+ \frac{x^3}{1\cdot 3}+ \frac{x^7}{1\cdot 3\cdot 7}+ \frac{x^{15}}{1\cdot 3\cdot 7\cdot 15}+ \frac{x^{31}}{1\cdot 3\cdot 7 \cdot 15\cdot 31}+\dots \end{align}