A systematic framework for designing second-order linear ordinary differential equation problems with controlled analytic complexity

Designing assessment problems for second-order linear ordinary differential equations (ODEs) often requires instructors to anticipate the analytic complexity of the solution process, particularly the integrals arising in general solution methods such as variation of parameters. However, this selection process is frequently based on intuition and trial-and-error rather than on a systematic construction procedure. This paper addresses this gap by presenting a systematic and reproducible framework for designing second-order linear ODE problems with controlled analytic complexity based on the structural form of the integrals arising in the variation of parameters formula. By prescribing the characteristic roots and selecting suitable auxiliary functions, the framework enables instructors to determine in advance the integrals involved in the solution process. The proposed approach allows the construction of problems with analytically tractable solution procedures, predetermined structural features, and explicit control over the computational demands encountered during the solution process.