I have a nonlinear system of ODEs of the form: $$ \dot{k} = k(t)^α-gk(t)-c(t)\\ \dot{c} = \frac{\alpha k(t)^{\alpha-1}-g-\rho + (1-\gamma)\frac{g}{A_0+gt}}{\gamma}c(t) $$ with associated initial condition $k(0)=k_0$ and terminal condition given by the transversality condition: $\lim_{t\to\infty} e^{-\rho t+(1-\gamma)\log(A(0)+gt)-\log(1-\gamma) }c(t)^{-\gamma}k(t)=0.$ In practice, I take the transversality condition to mean $\lim_{t\to\infty} k(t)=K$, where $K$ is found to be: $$K=(\frac{g+\rho}{\alpha})^{\frac{1}{\alpha-1}}$$ and $\lim_{t\to\infty} c(t) = C$, where $C$ is found to be: $$ C= K^α - gK. $$
I would like to check that the numerical solution I get from a boundary method bvp5c in Matlab is reasonable.
I had planned to use the method of matched asymptotics, expanding around $t=0$ and $t=\infty$, which would replace $\frac{g}{A_0+gt}$ with $\frac{g}{A_0}$ and $0$, but unfortunately have been unsuccessful so far in an analytical solution near these points. It is not obvious to me what a good 'semi-analytical' solution should be. Any suggestions on how I might get a good approximation to the solution would be greatly appreciated.
I should add there is a non-negativity constraint on k,c such that $k(t)\geq0, c(t)\geq0$ with $g,\rho,\gamma,\alpha>0$.