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Official problem 1 · A repeated-root differential equation
- Source locator
- Section A, official problem 1, paper PDF page 2
- Review note
- Independently derived; differential equation and initial data checked; official-key match · 2026-07-28
Problem summary
Solve a forced constant-coefficient second-order initial-value problem and evaluate the resulting solution at the requested point.
Core idea
Factor the differential operator, include a constant particular solution, and use the two initial conditions to determine both repeated-root coefficients.
Derivation
- 1Rewrite the operator as the square of D minus 1 and choose the constant particular solution 4.
- 2The repeated characteristic root contributes both exp(x) and x exp(x), so the general solution is 4 plus (C1 + C2 x) exp(x).
- 3The value condition gives C1 = 1.
- 4Differentiate once; the derivative condition then gives C2 = 1.
- 5Substituting x = 1 gives 4 + 2e.
Final answer
The requested value is 4 + 2e.