ODE of the form $y'=\exp(-(\cos(2\pi y))$

Question

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)?

Answer 1

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.

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, which strongly suggests that none of your ODE's has an explicit series approximation to its solutions.)

Answer 2

It can fall under " nonlinear Volterra integral equations":

$$u(x)=\int_{0}^{x} K(x,t)F(u(t))dt,$$

and so many approaches are possible:

• successive approximation
• series approximation as suggested from the other answer.
• Adomian Decomposition Method

See that and many other methods in the wonderful reference: "Linear and nonlinear integral equations"