Analytic solutions to nonlinear ODEs via spectral power series

Solutions to most nonlinear ordinary differential equations (ODEs) rely on numerical solvers, but this gives little insight into the nature of the trajectories and is relatively expensive to compute. In this paper, we derive analytic solutions to a class of nonlinear, homogeneous ODEs with linear and quadratic terms on the right-hand side. We formulate a power series expansion of each state variable, whose function depends on the eigenvalues of the linearized system, and solve for the coefficients using some linear algebra and combinatorics. Various experiments exhibit quickly decaying coefficients, such that a good approximation to the true solution consists of just a few terms.