An exact analytical solution is obtained for coupled time-independent Schrödinger equations with model potentials: , and . This is made possible by solving a single fourth-order ordinary differential equation derived from the original coupled equations. Exact closed-form solutions for the non-adiabatic transition matrices (or scattering matrices) are found for scattering boundary conditions with three, two and one open channels, respectively. How to apply the present results to deal with general potentials is also briefly analysed.