I have the following system of ODEs
$$\begin{aligned} \dot x (t) & = x(t) \frac{h(t)}{h_0} - \frac{x^2(t)}{x_0} \\ \dot h(t) & = -a h(t) \end{aligned}$$
where $x(0)=x_0$ and $h(0)=h_0$.
Using Mathematica, I have found that the system yields a complicated expression for x(t) that involves the exponential integral function. In contrast, $h(t)$ is simply $h(t)=h_0e^{-at}$.
Through trial and error, I have found that, when $a$ is small, $x(t) \approx x_0e^{-at}$. However, the accuracy of this approximation decreases as $t$ increases.
I am hoping to receive suggestions regarding an analytical approximation of this system that is fairly accurate but also integrable. Any help would be greatly appreciated!
I suppose that the solution you find complicated is $$x(t)=\frac {a x_0 e^{\frac{1-e^{-a t}}{a}} } {a+e^{\frac{1}{a}} \left(\text{Ei}\left(-\frac{1}{a}\right)-\text{Ei}\left(-\frac{e^{-a t}}{a}\right)\right) }$$
Assuming $t>0$ and $a>0$, you can expand as a series around $a=0$ to get $$x(t)=x_0+ x_0 \left(1-t-e^{-t}\right)\,a+\frac{1}{2} x_0 e^{-2 t} \left(-e^t t^2+e^{2 t} t^2+4 e^t t-2 e^{2 t}+2\right)a^2+O\left(a^3\right)$$
The syntax is
Now,
$$f(y)=\int_0^y x(t)\,dt=x_0 \log\Bigg[1+\frac{e^{\frac{1}{a}}}{a} \left(\text{Ei}\left(-\frac{1}{a}\right)-\text{Ei}\left(-\frac{e^{-a y}}{a}\right)\right)\Bigg]$$ which does not converge if $y\to \infty$.
If $a$ is very small, $f(y) \sim x_0y$