Algorithm for finding reduction transformation that transforms a rational ODE to solvable structure
Bibliographic record
Abstract
The integrability problem of rational first order ODE y'=M(x,y)/N(x,y), M,N in R[x,y], which can be viewed as a planar vector field, has been a long term research interest in area of dynamical system, physics etc. Since the invention of the computer algebra system, many algorithms have been developed to find the explicit first integral of such ODE and Maple has the most advanced function, called dsolve to handle this problem. In this paper, we would like to introduce an efficient algorithm which can be viewed as a supplement of the existing dsolve command, to judge whether a given rational ODE is of "reducible" structure, that is, via a rational, non-linear algebraic transformation in terms of the variable y, the ODE can be reduced to a more simple structure y'=Σni=0 fi(x)yi. Many of the known solvable ODE, including Bernoulli, Riccati, Abel, Chini are of this structure, and the integrable type of such ODE can be classified using invariants under linear transformation. We have implemented this algorithm based on Maple. And the overall performance and efficiency of this algorithm is considerable.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.017 | 0.005 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".