I want to solve the following equation for $f(x)$ (where $f: \mathbb{R}\mapsto\mathbb{R}^+$): $$ f(x)=\exp{\left(-\int_{-x}^{\infty}(y+x)^rf(y)\,dy\right)}\,\,, $$ where $r\in\mathbb{N}$ and $f(-\infty)=1$ and $f(+\infty)=0$. I am interested in particular in the cases $r=1$ and $r=2$.
I can manipulate the equation by defining an auxiliary function $a(x)$: $$ a(x)\equiv \ln{f(x)}\,,\quad \text{where} \quad a(-\infty)=0\,; $$ deriving both sides I get: $$ a'(x)=-r\int_{-x}^{\infty}dy(y+x)^{(r-1)} e^{a(y)}\,. $$
Case $r=1$:
Deriving another time both sides I get the following second order non-linear differential equation: $$ a''(x)=-e^{a(-x)}\,. $$ Defining $b(x)\equiv a(-x)$, it is possible to write a system of two differential equations $$ \left\{\begin{array}{ll} a''(x)=-e^{b(x)} & \\ b''(x)=-e^{a(x)} & \end{array}\right. $$ where some conditions must be imposed: $a(0)=b(0)$, $a(-\infty)=0$, $a'(-\infty)=0$ and $-a'(0)=b'(0)$.
Case $r=2$:
In this case I derive both sides twice to get: $$ a'''(x)=-2e^{a(-x)}\,. $$ In the same spirit of the $r=1$ case we can use $b(x)$ to write: $$ \left\{\begin{array}{ll} a'''(x)=-2e^{b(x)} & \\ b'''(x)=2e^{a(x)} & \end{array}\right. $$ where now the conditions to be imposed are: $a(0)=b(0)$, $a(-\infty)=0$, $a'(-\infty)=0$, $a''(-\infty)=0$, $-a'(0)=b'(0)$ and $a''(0)=b''(0)$.
This idea to manipulate these equations can be generalized to arbitrary $r$ and it is due to @yarchik from this question about numerical solutions. Is there a way to analytically deal with these equations?
The present question is also related to this one, which is the case $r=0$, that is analytically solvable.