Consider the function $h:[0,1]\to \mathbb{R}$
$$h(\theta):=\sum\_{k\geq 1}\frac{a\_{k}}{\sqrt{k}}\cos(2\pi k \theta)+\frac{b\_{k}}{\sqrt{k}}\sin(2\pi k \theta),$$
where $a\_{k},b\_{k}\in\mathbb{R}$. For simplicity let's take $\big(\frac{a\_{k}}{\sqrt{k}}\big),\big(\frac{b\_{k}}{\sqrt{k}}\big)\in \ell^{1}$, so that $h\in H^{1}(S^{1})$. 


The ODE we have is


$$y'=\exp(-h(y))\;\Leftrightarrow \;y(x)=\int\_{0}^{x}\exp(-h(y(t)))dt.$$


Does this fit anywhere into the ODE theory (e.g. fixed points of integral equations)? Maybe the simpler problem with a truncated series 


$$y'=\exp(-h^{N}(y)),$$


where $h^{N}(\theta)=\sum\_{k=1}^{N}\frac{a\_{k}}{\sqrt{k}}\cos(2\pi k \theta)+\frac{b\_{k}}{\sqrt{k}}\sin(2\pi k \theta)$. Or even simpler with a single term:


$$y'=\exp\Big(-\big(\frac{a\_{k}}{\sqrt{k}}\cos(2\pi k y)+\frac{b\_{k}}{\sqrt{k}}\sin(2\pi k y)\big)\Big).$$


Or even more simply


$$y'=\exp(-(\cos(2\pi y)).$$


Our main question is to obtain a (series) approximation solution to $y$ (for any of the above ODEs)?



ODE of the form $y'=\exp(-(\cos(2\pi y))$
If $f$ is a smooth enough function, then higher-order derivatives of a solution $y$ of the ODE 
\begin{equation}
 y'=f(y)
\end{equation}
can be found by successive differentiation of both sides of the ODE, giving $y''=f'(y)y'$ and, more generally, a recursion of the form 
\begin{equation}
 y^{(n)}=f\_n(y,y',\dots,y^{(n-1)})
\end{equation}
for natural $n$, where $f\_n$ is a certain function, depending on $f$, which can be given an explicit, albeit complicated, expression -- writing 
\begin{equation}
 f\_n(y,y',\dots,y^{(n-1)})=(f\circ y)^{(n-1)}
\end{equation}
and then using the [Faà di Bruno formula](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=2ahUKEwj8mrXqyJrgAhXE6oMKHc3jDDkQFjAAegQIChAC&url=https%3A%2F%2Fwww.ems-ph.org%2Fjournals%2Fshow_pdf.php%3Fissn%3D0013-6018%26vol%3D60%26iss%3D1%26rank%3D5&usg=AOvVaw0sDu0CdWeTiDK0F2BGpsZk). 


So, given an initial value $y\_0=y(t\_0)$, one obtains a Taylor approximation to $y$ in a neighborhood of $t\_0$. E.g., rescaling the argument of $y$, rewrite your last display as 
\begin{equation}
 y'=e^{-\cos y}. 
\end{equation}
Then, doing as described above and assuming $y(0)=0$, we get 
\begin{equation}
 y(t)=\frac1{1!e}\,t+\frac1{3!e^3}\,t^3+\frac6{5!e^5}\,t^5+\frac{87}{7!e^7}\,t^7+\cdots
\end{equation}
for $t$ near $0$. (The sequence of the numerators here does not match any of the sequences in [The On-Line Encyclopedia of Integer Sequences](https://oeis.org/search?q=1%2C+1%2C+6%2C+87%2C+2277%2C+92664%2C+5344569&language=english&go=Search), which strongly suggests that none of your ODE's has an explicit series approximation to its solutions.)


