# Question 2258: A Second-Order Candidate with Two Initial Data [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-0083) 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. R0003-02258 · q-20261013-advance-20261021-r04-initial-value-problem-0083 ## Original question and givens Use the integer polynomial candidate with ascending-power coefficients=[-5, -3, 0]. 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 a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=0; input grid=[-1, 0, 1], x0=0, required y0=-5, required y_prime0=-3. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?). - G001 — Use the integer polynomial candidate with ascending-power coefficients=[-5, -3, 0]. 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 a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=0; input grid=[-1, 0, 1], x0=0, required y0=-5, required y_prime0=-3. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?). ## 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=[-5, -3, 0] After initial point=0; required y₀=-5 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 · calculationBefore y′ coefficient at degree1 := mul(2, 0) After y′ coefficient at degree1 = 0 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, S002 S004 · calculationBefore y″ coefficient at degree0 := mul(1, 0) After y″ coefficient at degree0 = 0 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, S003 S005 · impliesBefore initial y ascending coefficients=[-5, -3, 0] 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, S004 S006 · calculationBefore initial y Horner multiply before degree0 := mul(-3, 0) After initial y Horner multiply before degree0 = 0 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore initial y Horner add degree0 := sum(0, -5) After initial y Horner add degree0 = -5 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore initial y′ ascending coefficients=[-3, 0] 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, S007 S009 · impliesBefore y(0)=-5; required=-5 After value condition=Yes Conditions: none Compare the actual function value at the initial point,not another grid point. Step identifiers and dependenciesRule: initial_value_condition · Uses: G001, S008 S010 · impliesBefore y′(0)=-3; required=-3 After derivative condition=Yes Conditions: none Second-order initial data also require the derivative condition;the two conditions cannot replace one another. Step identifiers and dependenciesRule: initial_derivative_condition · Uses: G001, S009 S011 · 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, S010 S012 · impliesBefore y at-1 ascending coefficients=[-5, -3, 0] 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, S011 S013 · 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, S012 S014 · calculationBefore y at-1 Horner add degree0 := sum(3, -5) After y at-1 Horner add degree0 = -2 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore y′ at-1 ascending coefficients=[-3, 0] 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, S014 S016 · impliesBefore y″ at-1 ascending coefficients=[0] 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, S015 S017 · calculationBefore a·y″ at-1 := mul(1, 0) After a·y″ at-1 = 0 Conditions: none Multiply each derivative by its own equation coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore b·y′ at-1 := mul(0, -3) After b·y′ at-1 = 0 Conditions: none Multiply each derivative by its own equation coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore c·y at-1 := mul(0, -2) After c·y at-1 = 0 Conditions: none Multiply each derivative by its own equation coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore left side at-1 := sum(0, 0, 0) After left side at-1 = 0 Conditions: none First sum the complete left side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore residual at-1 := sub(0, 0) After residual at-1 = 0 Conditions: none Define the residual as left side minus right side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · 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, S021 S023 · impliesBefore y at0 ascending coefficients=[-5, -3, 0] 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, S022 S024 · 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, S023 S025 · calculationBefore y at0 Horner add degree0 := sum(0, -5) After y at0 Horner add degree0 = -5 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore y′ at0 ascending coefficients=[-3, 0] 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, S025 S027 · impliesBefore y″ at0 ascending coefficients=[0] 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, S026 S028 · calculationBefore a·y″ at0 := mul(1, 0) After a·y″ at0 = 0 Conditions: none Multiply each derivative by its own equation coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore b·y′ at0 := mul(0, -3) After b·y′ at0 = 0 Conditions: none Multiply each derivative by its own equation coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore c·y at0 := mul(0, -5) After c·y at0 = 0 Conditions: none Multiply each derivative by its own equation coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore left side at0 := sum(0, 0, 0) After left side at0 = 0 Conditions: none First sum the complete left side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore residual at0 := sub(0, 0) After residual at0 = 0 Conditions: none Define the residual as left side minus right side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · 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, S032 S034 · impliesBefore y at1 ascending coefficients=[-5, -3, 0] 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, S033 S035 · 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, S034 S036 · calculationBefore y at1 Horner add degree0 := sum(-3, -5) After y at1 Horner add degree0 = -8 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · impliesBefore y′ at1 ascending coefficients=[-3, 0] 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, S036 S038 · impliesBefore y″ at1 ascending coefficients=[0] 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, S037 S039 · calculationBefore a·y″ at1 := mul(1, 0) After a·y″ at1 = 0 Conditions: none Multiply each derivative by its own equation coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore b·y′ at1 := mul(0, -3) After b·y′ at1 = 0 Conditions: none Multiply each derivative by its own equation coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore c·y at1 := mul(0, -8) After c·y at1 = 0 Conditions: none Multiply each derivative by its own equation coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore left side at1 := sum(0, 0, 0) After left side at1 = 0 Conditions: none First sum the complete left side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore residual at1 := sub(0, 0) After residual at1 = 0 Conditions: none Define the residual as left side minus right side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · 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, S043 S045 · impliesBefore all listed grid points pass=Yes; initial conditions pass=Yes After combined finite check=Yes 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, S044 S046 · 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, S045 ## Reverse checks - S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Recompute residual at1 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute left side at1 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute c·y at1 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute b·y′ at1 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute a·y″ at1 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Recompute y at1 Horner add degree0 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute y at1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute residual at0 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute left side at0 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute c·y at0 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute b·y′ at0 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute a·y″ at0 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Recompute y at0 Horner add degree0 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute y at0 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Recompute residual at-1 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute left side at-1 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute c·y at-1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute b·y′ at-1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute a·y″ at-1 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 y at-1 Horner add degree0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute y at-1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute initial y Horner add degree0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute initial y Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute y″ coefficient at degree0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute y′ coefficient at degree1 with exact rational arithmetic, not rounded decimals. - 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 ([-5, -3, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, -2, -3, 0, 0, Yes], [0, -5, -3, 0, 0, Yes], [1, -8, -3, 0, 0, Yes]], 0, -5, -3, -5, -3, Yes, Yes, Yes) 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: f0e73774877d97e6aff9a90f3acc8d50aba4f0ad420479945bac1c3138d00246 - candidate record SHA-256: 8f227a5af7ce136d4df2ff339ab66ff113f42b3e04cf16a5b4fb9a6638bf7185 - 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-0083.json)