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R0003-02275 · q-20261013-advance-20261021-r04-initial-value-problem-0100
Use the integer polynomial candidate with ascending-power coefficients=[12, 0, 2]. 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=5; input grid=[-1, 0, 1], x0=2, required y0=21, required y_prime0=8. 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=[12, 0, 2]. 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=5; input grid=[-1, 0, 1], x0=2, required y0=21, required y_prime0=8. 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?).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.
Before
candidate polynomial ascending coefficients=[12, 0, 2]
After
initial point=2; required y₀=21
Conditions: none
Retain the polynomial and initial data;the residual grid need not contain the initial point,so evaluate it separately.
Rule: candidate_IVP_data · Uses: G001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
y′ coefficient at degree1 := mul(2, 2)
After
y′ coefficient at degree1 = 4
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
y″ coefficient at degree0 := mul(1, 4)
After
y″ coefficient at degree0 = 4
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
initial y ascending coefficients=[12, 0, 2]
After
evaluate at x=2
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.
Rule: polynomial_evaluation_order · Uses: G001, S004
Before
initial y Horner multiply before degree1 := mul(2, 2)
After
initial y Horner multiply before degree1 = 4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
initial y Horner add degree1 := sum(4, 0)
After
initial y Horner add degree1 = 4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
initial y Horner multiply before degree0 := mul(4, 2)
After
initial y Horner multiply before degree0 = 8
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
initial y Horner add degree0 := sum(8, 12)
After
initial y Horner add degree0 = 20
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
initial y′ ascending coefficients=[0, 4]
After
evaluate at x=2
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.
Rule: polynomial_evaluation_order · Uses: G001, S009
Before
initial y′ Horner multiply before degree0 := mul(4, 2)
After
initial y′ Horner multiply before degree0 = 8
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
initial y′ Horner add degree0 := sum(8, 0)
After
initial y′ Horner add degree0 = 8
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
y(2)=20; required=21
After
value condition=No
Conditions: none
Compare the actual function value at the initial point,not another grid point.
Rule: initial_value_condition · Uses: G001, S012
Before
y′(2)=8; required=8
After
derivative condition=Yes
Conditions: none
Second-order initial data also require the derivative condition;the two conditions cannot replace one another.
Rule: initial_derivative_condition · Uses: G001, S013
Before
[-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.
Rule: finite_grid_domain · Uses: G001, S014
Before
y at-1 ascending coefficients=[12, 0, 2]
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.
Rule: polynomial_evaluation_order · Uses: G001, S015
Before
y at-1 Horner multiply before degree1 := mul(2, -1)
After
y at-1 Horner multiply before degree1 = -2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
y at-1 Horner add degree1 := sum(-2, 0)
After
y at-1 Horner add degree1 = -2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
y at-1 Horner multiply before degree0 := mul(-2, -1)
After
y at-1 Horner multiply before degree0 = 2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
y at-1 Horner add degree0 := sum(2, 12)
After
y at-1 Horner add degree0 = 14
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
y′ at-1 ascending coefficients=[0, 4]
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.
Rule: polynomial_evaluation_order · Uses: G001, S020
Before
y′ at-1 Horner multiply before degree0 := mul(4, -1)
After
y′ at-1 Horner multiply before degree0 = -4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
y′ at-1 Horner add degree0 := sum(-4, 0)
After
y′ at-1 Horner add degree0 = -4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
y″ at-1 ascending coefficients=[4]
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.
Rule: polynomial_evaluation_order · Uses: G001, S023
Before
a·y″ at-1 := mul(1, 4)
After
a·y″ at-1 = 4
Conditions: none
Multiply each derivative by its own equation coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
b·y′ at-1 := mul(0, -4)
After
b·y′ at-1 = 0
Conditions: none
Multiply each derivative by its own equation coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
c·y at-1 := mul(0, 14)
After
c·y at-1 = 0
Conditions: none
Multiply each derivative by its own equation coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
left side at-1 := sum(4, 0, 0)
After
left side at-1 = 4
Conditions: none
First sum the complete left side.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
residual at-1 := sub(4, 5)
After
residual at-1 = -1
Conditions: none
Define the residual as left side minus right side.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
residual at-1=-1
After
point equation match=No
Conditions: none
This flag answers only for the current grid point.
Rule: point_equation_check · Uses: G001, S029
Before
y at0 ascending coefficients=[12, 0, 2]
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.
Rule: polynomial_evaluation_order · Uses: G001, S030
Before
y at0 Horner multiply before degree1 := mul(2, 0)
After
y at0 Horner multiply before degree1 = 0
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
y at0 Horner add degree1 := sum(0, 0)
After
y at0 Horner add degree1 = 0
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
y at0 Horner multiply before degree0 := mul(0, 0)
After
y at0 Horner multiply before degree0 = 0
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
y at0 Horner add degree0 := sum(0, 12)
After
y at0 Horner add degree0 = 12
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
y′ at0 ascending coefficients=[0, 4]
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.
Rule: polynomial_evaluation_order · Uses: G001, S035
Before
y′ at0 Horner multiply before degree0 := mul(4, 0)
After
y′ at0 Horner multiply before degree0 = 0
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
y′ at0 Horner add degree0 := sum(0, 0)
After
y′ at0 Horner add degree0 = 0
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
y″ at0 ascending coefficients=[4]
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.
Rule: polynomial_evaluation_order · Uses: G001, S038
Before
a·y″ at0 := mul(1, 4)
After
a·y″ at0 = 4
Conditions: none
Multiply each derivative by its own equation coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
b·y′ at0 := mul(0, 0)
After
b·y′ at0 = 0
Conditions: none
Multiply each derivative by its own equation coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
c·y at0 := mul(0, 12)
After
c·y at0 = 0
Conditions: none
Multiply each derivative by its own equation coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
left side at0 := sum(4, 0, 0)
After
left side at0 = 4
Conditions: none
First sum the complete left side.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
residual at0 := sub(4, 5)
After
residual at0 = -1
Conditions: none
Define the residual as left side minus right side.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
residual at0=-1
After
point equation match=No
Conditions: none
This flag answers only for the current grid point.
Rule: point_equation_check · Uses: G001, S044
Before
y at1 ascending coefficients=[12, 0, 2]
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.
Rule: polynomial_evaluation_order · Uses: G001, S045
Before
y at1 Horner multiply before degree1 := mul(2, 1)
After
y at1 Horner multiply before degree1 = 2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
y at1 Horner add degree1 := sum(2, 0)
After
y at1 Horner add degree1 = 2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
y at1 Horner multiply before degree0 := mul(2, 1)
After
y at1 Horner multiply before degree0 = 2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
y at1 Horner add degree0 := sum(2, 12)
After
y at1 Horner add degree0 = 14
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
y′ at1 ascending coefficients=[0, 4]
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.
Rule: polynomial_evaluation_order · Uses: G001, S050
Before
y′ at1 Horner multiply before degree0 := mul(4, 1)
After
y′ at1 Horner multiply before degree0 = 4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
y′ at1 Horner add degree0 := sum(4, 0)
After
y′ at1 Horner add degree0 = 4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
y″ at1 ascending coefficients=[4]
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.
Rule: polynomial_evaluation_order · Uses: G001, S053
Before
a·y″ at1 := mul(1, 4)
After
a·y″ at1 = 4
Conditions: none
Multiply each derivative by its own equation coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
b·y′ at1 := mul(0, 4)
After
b·y′ at1 = 0
Conditions: none
Multiply each derivative by its own equation coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
c·y at1 := mul(0, 14)
After
c·y at1 = 0
Conditions: none
Multiply each derivative by its own equation coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
left side at1 := sum(4, 0, 0)
After
left side at1 = 4
Conditions: none
First sum the complete left side.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
residual at1 := sub(4, 5)
After
residual at1 = -1
Conditions: none
Define the residual as left side minus right side.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
residual at1=-1
After
point equation match=No
Conditions: none
This flag answers only for the current grid point.
Rule: point_equation_check · Uses: G001, S059
Before
all listed grid points pass=No; 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.
Rule: finite_grid_initial_conjunction · Uses: G001, S060
Before
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.
Rule: grid_vs_interval_scope · Uses: G001, S061
S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S059 · premise_and_result_audit — Recompute residual at1 with exact rational arithmetic, not rounded decimals.S058 · premise_and_result_audit — Recompute left side at1 with exact rational arithmetic, not rounded decimals.S057 · premise_and_result_audit — Recompute c·y at1 with exact rational arithmetic, not rounded decimals.S056 · premise_and_result_audit — Recompute b·y′ at1 with exact rational arithmetic, not rounded decimals.S055 · premise_and_result_audit — Recompute a·y″ at1 with exact rational arithmetic, not rounded decimals.S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S053 · premise_and_result_audit — Recompute y′ at1 Horner add degree0 with exact rational arithmetic, not rounded decimals.S052 · premise_and_result_audit — Recompute y′ at1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S050 · premise_and_result_audit — Recompute y at1 Horner add degree0 with exact rational arithmetic, not rounded decimals.S049 · premise_and_result_audit — Recompute y at1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S048 · premise_and_result_audit — Recompute y at1 Horner add degree1 with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Recompute y at1 Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.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 — Recompute residual at0 with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Recompute left side at0 with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute c·y at0 with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute b·y′ at0 with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Recompute a·y″ at0 with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S038 · premise_and_result_audit — Recompute y′ at0 Horner add degree0 with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute y′ at0 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S035 · premise_and_result_audit — Recompute y at0 Horner add degree0 with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute y at0 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Recompute y at0 Horner add degree1 with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute y at0 Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S029 · premise_and_result_audit — Recompute residual at-1 with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute left side at-1 with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute c·y at-1 with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute b·y′ at-1 with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute a·y″ at-1 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 — Recompute y′ at-1 Horner add degree0 with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute y′ at-1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S020 · premise_and_result_audit — Recompute y at-1 Horner add degree0 with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute y at-1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute y at-1 Horner add degree1 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute y at-1 Horner multiply before degree1 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Recompute initial y′ Horner add degree0 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute initial y′ Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.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 — Recompute initial y Horner add degree0 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute initial y Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute initial y Horner add degree1 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute initial y Horner multiply before degree1 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.([12, 0, 2], 1, 0, 0, 5, [-1, 0, 1], [[-1, 14, -4, 4, -1, No], [0, 12, 0, 4, -1, No], [1, 14, 4, 4, -1, No]], 2, 20, 8, 21, 8, No, Yes, No)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
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