Exact Solution of First-Order Ordinary Differential Equations by Multiple Integral Transform Methods: A Four-Parameter Unification of the Gbenga Gideon Integral Transforms with Numerical Validation

Linear first-order initial value problems are routinely solved in the literature by a growing catalogue of Laplace-type integral transforms, including the Laplace, Aboodh, Elzaki, Mahgoub, Mohand, and Sumudu transforms. A sizeable strand of recent work solves the same benchmark equations by two or more of these transforms and reports, example by example, that the answers coincide. What that strand has lacked is a structural explanation of the coincidence and an independent check of the computed solutions.
This study solves first-order ordinary differential equations with constant coefficients using the Gbenga Gideon integral transform (GGIT), a four-parameter Laplace-type transform with kernel pv^m e^(-qv^w t), and simultaneously by the Laplace, Aboodh, and Elzaki transforms. The central aim is to prove, not merely observe, that the four methods must return the same solution, and to quantify the exactness of that solution numerically.
The operational calculus of the GGIT needed for first-order problems is developed as formal theorems with complete proofs: transforms of elementary functions, linearity, both shifting theorems, change of scale, and transforms of derivatives. A parameter dictionary identifies the Laplace, Aboodh, and Elzaki transforms as fixed choices of (p,q,m,w), and a method-invariance proposition converts the empirical agreement of the four calculations into a corollary of uniqueness. Four benchmark initial value problems spanning unforced decay, polynomial forcing, harmonic forcing, and constant forcing are then solved by all four methods, and every closed-form solution is validated against adaptive Runge-Kutta integration at tolerance 10^(-10).
The four transform methods produce identical solutions for each benchmark, exactly as the invariance proposition requires. Across the four problems the maximum absolute discrepancy between the closed forms and the numerical reference lies between 2.4×10^(-11) and 1.0×10^(-9), with root-mean-square errors between 4.4×10^(-12) and 1.4×10^(-10), at the level of the integrator’s own tolerance. A symbolic benchmark shows the transform-domain route resolving the harmonically forced problem approximately 5.4 times faster than direct symbolic integration.
For first-order constant-coefficient problems, the choice among Laplace-type transforms is a choice of bookkeeping, not of mathematical power: all are instances of a single four-parameter calculus, and their agreement is a theorem. The GGIT makes this explicit by carrying the parameters symbolically through one computation that specializes to every named method at once.

first-order ordinary differential equation; integral transform; Gbenga Gedion integral transform

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