A-LevelFurther MathematicsDifferential equations9231/12 · Oct/Nov 2012 · Q36 Marks
Find the general solution of the differential equation d²x/dt² + 4dx/dt + 13x = 26t² + 3t + 13.
- Find the general solution of the differential equation d²x/dt² + 4dx/dt + 13x = 26t² + 3t + 13.[6]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2012). The mark or mark type is shown in brackets.
- [M1] Solves AQE: m² + 4m + 13 = 0 ⇒ m = −2 ±3i
- [A1] Finds CF: e⁻²ᵗ (Acos 3t + B sin 3t)
- [M1] Form for PI and differentiates: x = pt² + qt + r ⇒ x' = 2pt + q ⇒ x'' = 2p
- [M1A1] Compares coefficients and solves: 13p = 26 ⇒ p=2; 8p+13q = 3 ⇒ q = -1; 2p+4q+13r =13⇒r=1
- [A1] GS: x = e⁻²ᵗ (A cos 3t + B sin 3t) + 2t² − t + 1
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