A new approach to the solution of burnup equations is developed that takes into account the dependence of the reaction constants on time as well as nonlinear and feedback effects. With the help of the transition probabilities for the simplified problem, the burnup differential equation is reduced to the equivalent integral equation, which is solved by iterations. The solution is made easy to understand with the help of diagrams constructed following the suggested rules. It is strictly proved that any nuclide transmutation network can be broken into independent depletion chains if the burnup equations are linear in concentrations. The theory is illustrated by examples of the time dependence of reaction constants.