SIRPY — solutions that show their evidence

SIRPY solves differential and integral equations and delivers, with every answer, the measured evidence that it satisfies your equation and your conditions — and a plain statement of what was not checked. You receive the solution, a data artifact you can verify yourself with a hash check and ordinary Python, and a report at one of three levels:

Verified (measured facts and figures),

Certified Standard (four formal, measured claims, built for audit and compliance files), or

Certified Premium (full diagnostics and analytics).

See SIRPY in Action

See a complete three-level report set on Troesch’s problem — the one two independent reviewers passed — at

github.com/sirpy26/SIRPY/tree/main/Troesch, and more examples at

github.com/sirpy26/SIRPY.

Then send us your problem through the contact page: we’ll run it and tell you exactly what happened.

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The SIRPY Problem Library — Solved and Verified

Every solver claims accuracy. We publish evidence.

Each entry below is one real problem: the exact input we handed SIRPY, the exact output it returned, and an honest read of what happened — including the runs that failed, because how a solver behaves when it cannot solve something is part of what you are buying. Every number is checkable. Several problems have exact solutions you can verify by hand.

Entries 1–7 are written up in full on this site. Entries 8 onward — and every new problem from now on — live in our GitHub repository, which is updated frequently: GitHub link. If you want to see what SIRPY did this week, that is where to look.

Independently audited. In August 2026, auditor Alfred T. McBride reviewed SIRPY under his Boundary-Conditioned Reality framework on Problem 1 below — a nonlinear boundary value problem whose true solution blows up just past the edge of the region being solved. The ruling was a full pass for that benchmark. His standard is worth quoting: “a failed intermediate method is not hidden when a subsequent method achieves closure.” SIRPY passed partly because it reported a method that did not work before the one that did.

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1 · The Problem That Hides a Cliff

y″ = 2y³, y(0) = −0.1, y(9) = −1

Looks harmless. Its exact solution is y = 1/(x−10) — smooth on the whole interval, with a pole waiting one unit past the right boundary. A solver that isn’t paying attention returns a confident curve and never mentions the cliff. SIRPY recovered the exact solution — endpoint agreement at machine precision — and located the unseen singularity at x ≈ 9.977 without being told it existed. This is the problem the independent audit was run on. Full write-up on this site.

2 · The Solution You’ll Swear Is Fake

Same equation — one number changed: y(11) = 1.

Last time the pole sat at x = 10, just outside. Now the interval extends past it. A tool that solved Problem 1 by luck fails here by disaster. SIRPY identified that the blow-up now sits inside the interval and said so, instead of threading a plausible-looking curve through infinity. Two problems, one changed digit, opposite correct answers — that is what “reading the problem” means. Full write-up on this site.

3 · The Equation That Depends on Its Own Answer

y″ = −y ∫₀¹ y² dt, y(0) = 1, y(1) = 0

To evaluate the right-hand side at any single point you need the whole solution — the integral sees all of it at once. Tools that march point-by-point have nothing to march with. This structure is real physics: Gross–Pitaevskii, Hartree, population models where growth depends on total population. SIRPY solved it directly and verified the result against the equation as written — integral and all. Full write-up on this site.

4 · The Gross–Pitaevskii Equation

−y″ + x²y + y³ = μy, y(±5) = 0, ∫|y|² dt = 1

The standard model of a Bose–Einstein condensate — and here the solution, its slope, and the eigenvalue μ are all unknown at once, tied together by a normalization constraint. Three unknowns, one shot. SIRPY determined all three simultaneously and verified the triple against every condition. Full write-up on this site.

5a · The Equation That Tells You Where It Starts

y(t) = t(1−t) + ∫₀ᵗ (t−s) y(s)² ds

No derivatives anywhere — the unknown lives under the integral, and its value at each point depends on its own accumulated history, squared. Set t to the left endpoint and the equation supplies its own starting value; nothing is yours to choose. Memory materials, delayed-recruitment populations, renewal processes. Solved and verified as an integral equation, not converted into something easier. Full write-up on this site.

5b · A Boundary Condition Where None Can Exist

Same Volterra equation, plus a demand: y(1) = 0.

A Volterra equation has no free constants — its solution is already fully determined, so an extra condition is, as posed, impossible. The resolution is to make room deliberately: leave an unknown constant in the forcing and ask SIRPY to find the value for which the boundary condition holds. It did, turning an over-determined problem into a well-posed one and verifying both the equation and the added condition. Full write-up on this site.

6 · Troesch’s Problem — a Famous Benchmark, Zero Tuning, and an Honest Refusal

y″ = μ sinh(μy), y(0) = 0, y(1) = 1, at μ = 10 and 15

A classical plasma-physics benchmark built to punish solvers: the solution hugs zero, then climbs to 1 inside a razor-thin layer; at μ = 15 the initial slope is of order 10⁻⁶ and errors amplify exponentially. We ran it with no tuning at all — and this entry includes a run that failed, published, because the refusal came with its reason and its remedy rather than a wrong answer. Full write-up on this site.

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The remaining entries are summarized here; the full write-ups, with complete inputs and outputs, are on GitHub — along with every new problem as it lands: GitHub link.

7 · Where the Guarantee Ends — a Stiff Classic

y″ + 101y′ + 100y = 0, y(0) = 1, y′(0) = −1

The exact solution is just e⁻ˣ — but the equation carries a hidden fast mode, e⁻¹⁰⁰ˣ, absent from the answer and hostile to any method that ignores it. We entered it three ways, escalating the demand each time, and mapped exactly where the guarantee extends and where it ends — with the boundary stated, not discovered by the client.

8 · The Bratu Problem Near Criticality

y″ + λeʸ = 0, y(0) = y(1) = 0, near λc ≈ 3.513830719

Below the critical value, two solutions; at it, exactly one; above, none. We drove SIRPY toward the fold and documented what breaks, what survives, and which apparent effects are numerical artifacts rather than mathematics — a case study in not fooling yourself near a bifurcation.

9 · Thomas–Fermi: Singular Origin, Infinite Domain

The 1927 statistical model of the multi-electron atom: no closed form exists, the equation is singular at the origin, and the boundary condition sits at infinity. SIRPY recovered the famous initial slope y′(0) ≈ −1.5880710226 — the constant tabulated in the atomic-physics literature — with verification at every step.

10 · The Blasius Boundary Layer

The flat-plate boundary layer of fluid mechanics, posed on an infinite domain. SIRPY recovered the classical Blasius constant (≈ 0.4696) and — unusually — reported a truncation certificate: a measured statement of how much the artificial far boundary affects the answer, instead of a silent hope that it doesn’t.

11 · From Blasius to Falkner–Skan

One equation, a family of flows — accelerating, decelerating, wedge angles between. The entire family solved from one four-line setup, each member verified, showing what it looks like when a solver treats a class of problems rather than one.

12 · The Sturm–Liouville Eigenvalue Problem

y″ = −λy, y(0) = y(1) = 0

Every requirement is satisfied by y ≡ 0, for every λ — the problem has a hole in it, and most tools fall straight through, returning zero. SIRPY noticed the hole, excluded the trivial solution, and recovered the eigenvalue λ = π² to eight digits from the boundary conditions alone.

13 · An Integro-Differential Equation with an Oscillatory Kernel

A differential equation coupled to its own oscillating history — the kind of memory term that appears in viscoelasticity and wave problems, and that smears error across the whole interval if mishandled. Solved as 46 analytic pieces with the equation, history term included, verified by substitution.

14 · Certificate of Nonexistence

y″ + 3.52 eʸ = 0, y(0) = y(1) = 0 — just past the fold

Here the correct answer is that no solution exists, and SIRPY said so with evidence rather than an error message: candidates examined, range searched, the closest any came to meeting the boundary condition, and the largest interval that would be solvable (0.999123, matching the exact 0.999123297). Most tools cannot distinguish “impossible” from “I failed.” This one can, and shows its work.

15 · The Fourth-Order Bratu Problem

y⁗ = λeʸ — a beam under a load that grows with deflection

The critical loads for two standard supports, each obtained twice by methods sharing neither discretisation nor algorithm: λc = 34.6402 (simply supported) and 176.4714 (clamped), agreeing across routes to 0.03 % and 0.004 %. Both solution branches traced, including the one no straight-ahead method can reach.

16 · HIRES — Eight Coupled Equations from Photochemistry

A standard stiff benchmark: eight nonlinear equations, rate constants spanning five orders of magnitude. SIRPY solved and verified the system — and detected, unprompted, that y₇ + y₈ is conserved, then measured its drift at 6×10⁻¹⁸. A quantity the solver was never told about, held to eighteen digits, is evidence no residual can fake.

17 · Van der Pol Across Three Regimes

y″ = μ(1−y²)y′ − y at μ = 10, 100, 1000

One line changed across a hundredfold change in stiffness. At μ = 1000 SIRPY diagnosed the cost, forecast the bill before paying it, handed the problem to a specialist routine with explicit credit — then audited the borrowed answer to 7×10⁻¹¹. Asked to do it the long way anyway, it did: thousands of exact analytic pieces, every joint measured.

18 · Lane–Emden — Starting Where the Equation Breaks

y″ = −2y′/x − y⁵, y(0) = 1, y′(0) = 0

The initial data sit exactly where the equation divides by zero — the centre of the star. SIRPY started at the singular point, selected the physically admissible solution and said why, and estimated the location of trouble spots that sit off the real axis entirely: 1.75 against the true √3 ≈ 1.732, from real data alone. It also caught one of its own diagnostics raising a false alarm — and overruled it, with the reasoning printed.

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What happens next

New problems are added regularly — on GitHub. Star the repository to see each one as it lands: GitHub

Have a question about any entry? Our site assistant can walk you through what each problem shows and what the numbers mean. And if you have a problem of your own — one of these, or one of yours — send it. We’ll run it and tell you exactly what happened, including if we couldn’t do it.

If a result cannot be verified, we say so — and you don’t pay for it.

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The independent audit