# Question 2222: A First-Order Candidate: Grid Residuals and Initial Value [Return to catalogue page 12](https://ocme.evemisslab.com/en/questions/page/12/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261021-r04-initial-value-problem-foundations/#q-20261013-advance-20261021-r04-initial-value-problem-0072) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0003-02222 · q-20261013-advance-20261021-r04-initial-value-problem-0072 ## Original question and givens Use the integer polynomial candidate with ascending-power coefficients=[9, -3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-3; input grid=[-1, 0, 1], x0=-1, required y0=13. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?). - G001 — Use the integer polynomial candidate with ascending-power coefficients=[9, -3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-3; input grid=[-1, 0, 1], x0=-1, required y0=13. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-initial-value-problem](https://ocme.evemisslab.com/en/mko/mko-initial-value-problem/) ## Step-by-step derivation S001 · impliesBefore candidate polynomial ascending coefficients=[9, -3] After initial point=-1; required y₀=13 Conditions: none Retain the polynomial and initial data;the residual grid need not contain the initial point,so evaluate it separately. Step identifiers and dependenciesRule: candidate_IVP_data · Uses: G001 S002 · calculationBefore y′ coefficient at degree0 := mul(1, -3) After y′ coefficient at degree0 = -3 Conditions: none Multiply by the original degree and lower it by1;the constant term differentiates to0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore initial y ascending coefficients=[9, -3] After evaluate at x=-1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S002 S004 · calculationBefore initial y Horner multiply before degree0 := mul(-3, -1) After initial y Horner multiply before degree0 = 3 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore initial y Horner add degree0 := sum(3, 9) After initial y Horner add degree0 = 12 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore y(-1)=12; required=13 After value condition=No Conditions: none Compare the actual function value at the initial point,not another grid point. Step identifiers and dependenciesRule: initial_value_condition · Uses: G001, S005 S007 · impliesBefore [-1, 0, 1] After distinct ordered audit grid=[-1, 0, 1] Conditions: none Deduplication and sorting fix presentation;every listed distinct point is then audited. Step identifiers and dependenciesRule: finite_grid_domain · Uses: G001, S006 S008 · impliesBefore y at-1 ascending coefficients=[9, -3] After evaluate at x=-1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S007 S009 · calculationBefore y at-1 Horner multiply before degree0 := mul(-3, -1) After y at-1 Horner multiply before degree0 = 3 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore y at-1 Horner add degree0 := sum(3, 9) After y at-1 Horner add degree0 = 12 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore y′ at-1 ascending coefficients=[-3] After evaluate at x=-1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S010 S012 · calculationBefore p·y at-1 := mul(0, 12) After p·y at-1 = 0 Conditions: none Compute the p-scaled function value on the right. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore y′−p·y at-1 := sub(-3, 0) After y′−p·y at-1 = -3 Conditions: none Retain the correct subtraction order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore residual at-1 := sub(-3, -3) After residual at-1 = 0 Conditions: none Subtract q to obtain the full residual. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore residual at-1=0 After point equation match=Yes Conditions: none This flag answers only for the current grid point. Step identifiers and dependenciesRule: point_equation_check · Uses: G001, S014 S016 · impliesBefore y at0 ascending coefficients=[9, -3] After evaluate at x=0 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S015 S017 · calculationBefore y at0 Horner multiply before degree0 := mul(-3, 0) After y at0 Horner multiply before degree0 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore y at0 Horner add degree0 := sum(0, 9) After y at0 Horner add degree0 = 9 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore y′ at0 ascending coefficients=[-3] After evaluate at x=0 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S018 S020 · calculationBefore p·y at0 := mul(0, 9) After p·y at0 = 0 Conditions: none Compute the p-scaled function value on the right. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore y′−p·y at0 := sub(-3, 0) After y′−p·y at0 = -3 Conditions: none Retain the correct subtraction order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore residual at0 := sub(-3, -3) After residual at0 = 0 Conditions: none Subtract q to obtain the full residual. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore residual at0=0 After point equation match=Yes Conditions: none This flag answers only for the current grid point. Step identifiers and dependenciesRule: point_equation_check · Uses: G001, S022 S024 · impliesBefore y at1 ascending coefficients=[9, -3] After evaluate at x=1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S023 S025 · calculationBefore y at1 Horner multiply before degree0 := mul(-3, 1) After y at1 Horner multiply before degree0 = -3 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore y at1 Horner add degree0 := sum(-3, 9) After y at1 Horner add degree0 = 6 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · impliesBefore y′ at1 ascending coefficients=[-3] After evaluate at x=1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S026 S028 · calculationBefore p·y at1 := mul(0, 6) After p·y at1 = 0 Conditions: none Compute the p-scaled function value on the right. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore y′−p·y at1 := sub(-3, 0) After y′−p·y at1 = -3 Conditions: none Retain the correct subtraction order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore residual at1 := sub(-3, -3) After residual at1 = 0 Conditions: none Subtract q to obtain the full residual. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · impliesBefore residual at1=0 After point equation match=Yes Conditions: none This flag answers only for the current grid point. Step identifiers and dependenciesRule: point_equation_check · Uses: G001, S030 S032 · impliesBefore all listed grid points pass=Yes; initial conditions pass=No After combined finite check=No Conditions: none Every requested grid and initial condition must pass;do not omit a point or an initial condition. Step identifiers and dependenciesRule: finite_grid_initial_conjunction · Uses: G001, S031 S033 · impliesBefore finite grid agreement alone is not a general interval certificate After a polynomial coefficient/degree certificate is a separate possible strengthening Conditions: none For these polynomials,coefficient identity or a proved degree bound with enough distinct points could separately strengthen the evidence;this output flag records only the finite check,not a denial of that additional method. Step identifiers and dependenciesRule: grid_vs_interval_scope · Uses: G001, S032 ## Reverse checks - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Recompute residual at1 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute y′−p·y at1 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute p·y at1 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Recompute y at1 Horner add degree0 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute y at1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Recompute residual at0 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute y′−p·y at0 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute p·y at0 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Recompute y at0 Horner add degree0 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute y at0 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Recompute residual at-1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute y′−p·y at-1 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute p·y at-1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute y at-1 Horner add degree0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute y at-1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute initial y Horner add degree0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute initial y Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Recompute y′ coefficient at degree0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([9, -3], 0, -3, [-1, 0, 1], [[-1, 12, -3, 0, Yes], [0, 9, -3, 0, Yes], [1, 6, -3, 0, Yes]], -1, 12, 13, No, No) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: ff44c0e39de8245792be3a6541013cde57e7112f941bf70b067eced37405f9da - candidate record SHA-256: f6af315c9aa870ab69d05279f71d8fc71fc0018642496f9027910463d2019f8e - semantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783f - question-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021fa - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261021-r04-initial-value-problem-0072.json)