Changed equations, inexact iterates and continuation
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 150 minutes, as an unvalidated planning estimate. This collection contains 3 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Name the function space, rectangle and interval before applying the contraction theorem.
Distinguish exact integral identities, numerical approximations and uniform error certificates.
A failed sufficient hypothesis alone proves neither nonexistence nor nonuniqueness.
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For y′=2y+t, y(0)=0, use the rectangle |t|≤1/4, |y|≤1. Construct the integral operator and prove a unique solution in its closed function ball. Starting from u₀=0, compute u₁ and u₂. Give an a priori sup-norm error bound for uₙ. Then verify an explicit solution and justify uniqueness on every finite interval, rather than claiming the first contraction estimate automatically applies on the whole line.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let T be the Picard operator for y′=1+y² on I=[−1/4,1/4], with ball X={u:‖u‖∞≤1} and contraction factor q=1/2. A computation produces continuous functions vₙ∈X, v₀=0, and rigorously established uniform defects ‖vₙ−Tvₙ₋₁‖∞≤δₙ. Prove an error estimate against the exact fixed point y. Evaluate it first for δₙ=10⁻²·4⁻ⁿ, then for δₙ=10⁻³. Does a nonzero constant error allowance prove convergence fails? Explain what must be added if the only evidence is a table of sampled defects or if an iterate leaves X.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For y′=exp(y), y(0)=0, first construct a local solution using |t|≤1, |y|≤1 and h=1/(2e). Verify y(t)=−log(1−t) on t<1 and prove uniqueness wherever a solution exists with this initial value. Explain why the smooth vector field does not yield a finite real solution through t=1. Relate this failure to the compact-interior continuation condition, and determine whether a finite negative-time endpoint is possible.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
For y′=2y+t, y(0)=0, use the rectangle |t|≤1/4, |y|≤1. Construct the integral operator and prove a unique solution in its closed function ball. Starting from u₀=0, compute u₁ and u₂. Give an a priori sup-norm error bound for uₙ. Then verify an explicit solution and justify uniqueness on every finite interval, rather than claiming the first contraction estimate automatically applies on the whole line.
Let T be the Picard operator for y′=1+y² on I=[−1/4,1/4], with ball X={u:‖u‖∞≤1} and contraction factor q=1/2. A computation produces continuous functions vₙ∈X, v₀=0, and rigorously established uniform defects ‖vₙ−Tvₙ₋₁‖∞≤δₙ. Prove an error estimate against the exact fixed point y. Evaluate it first for δₙ=10⁻²·4⁻ⁿ, then for δₙ=10⁻³. Does a nonzero constant error allowance prove convergence fails? Explain what must be added if the only evidence is a table of sampled defects or if an iterate leaves X.
For y′=exp(y), y(0)=0, first construct a local solution using |t|≤1, |y|≤1 and h=1/(2e). Verify y(t)=−log(1−t) on t<1 and prove uniqueness wherever a solution exists with this initial value. Explain why the smooth vector field does not yield a finite real solution through t=1. Relate this failure to the compact-interior continuation condition, and determine whether a finite negative-time endpoint is possible.
