The Second-Order Basis for Homogeneous Solutions of Compatible Higher Order Linear ODEs with Varying Coefficients

Nugroho, Gunawan; Biyanto, Totok Ruki

doi:10.30511/ttmp.2026.2091664.1080

The Second-Order Basis for Homogeneous Solutions of Compatible Higher Order Linear ODEs with Varying Coefficients

Document Type : Original Article

Authors

Gunawan Nugroho ¹

Totok Ruki Biyanto ²

¹ JL. WONORUNGKUT UTARA I/23, RUNGKUT

² Department of Engineering Physics, Institut Teknologi Sepuluh Nopember

10.30511/ttmp.2026.2091664.1080

Abstract

This research presents a reduction and reconstruction study for third and fourth order linear ordinary differential equations with variable coefficients, including equations with nonzero dependent variable terms. A compatible higher order equation is reduced to a second order ODE and then is solved analytically to produce the homogeneous solution required by variation of parameters and the Wronskian. Once this homogeneous solution is found, the systematic general solution follows from the classical variation of parameters, represented here by equation (9d). The work therefore aims to pierce the main difficulty of variable coefficient problems: finding an analytical homogeneous solution for the linear compatible ODE with varying coefficients. Validation is performed on Airy, Bessel, Legendre, and Weber type models, including induced third and fourth order equations and beyond special functions coefficients. Numerical comparisons with classical special functions show agreement at round off scale.

Keywords

Variable-coefficient ordinary differential equations

reduction of order

analytical solution

integrating factor

special functions

Subjects