Coupled second order non linear ODE

Question

How can I solve two coupled non-linear second order ODE with following equation $$ y_1'' = e^{-k_1(y_1-y_2)}-e^{-k_2(y_2-y_1)} \\ y_2'' = e^{-k_1(y_1-y_2)}-e^{-k_2(y_2-y_1)} $$

Answer 1

As $y_1''=y_2''$ we get $y_1-y_2=ax+b$ and the differential equations reduce to $$ y_1''=e^{-k_1(ax+b)}-e^{k_2(ax+b)}\\ y_2''=e^{-k_1(ax+b)}-e^{k_2(ax+b)} $$ which can now easily be integrated twice.