The following system of two non-linear second order ODEs comes up in my research:
\begin{align*} & x''(t)+a_1 x'(t)+a_2 x(t)^\alpha+a_3\frac{1}{1+e^t}+a_4=0\\ & x(t)^\beta\Big(y''(t)+b_1 y'(t)+b_2\frac{1}{1+e^t}+b_3\Big)+b_4 y(t)+b_5 x(t)=0 \end{align*} subject to the following conditions: \begin{align*} &\lim_{t\to\infty} x(t)=\lim_{t\to\infty} y(t)=A\\ & x'(t_0)=y'(t_0)=0\\ & y(t_0)=y_0 \end{align*} where the coefficients $a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4,b_5$ and parameters $\alpha,\beta, y_0, A$ are given and $t_0$ is to be solved for. My questions are:
How do I prove the existence and uniqueness theorem for such a system?
How can I solve the system numerically? (I'm assuming an analytical solution does not exist) Please note that $t_0$ needs to be solved for as well.
Any suggestions will be greatly appreciated. Thank you!