Find the particular solution of the differential equation d²y/dx² + 2dy/dx + 5y = 10e⁻²ˣ, given that y = 5 and dy/dx = 1 when x = 0.

Mark scheme

Marking points from the official mark scheme (May/June 2012). The mark or mark type is shown in brackets.

1. [M1A1] Forms AQE and solves. m² + 2m + 5 = 0 ⇒ m = -1 ± 2i
2. [A1] Writes CF. CF: y = e⁻ˣ (A cos 2x + B sin 2x)
3. [M1] Correct form for PI and differentiates twice. y = ke⁻²ˣ ⇒ y' = -2ke⁻²ˣ ⇒ y'' = 4e⁻²ˣ
4. [M1] Substitutes. ⇒ 4k - 4k + 5k = 10 ⇒ k=2
5. [A1] Writes PI. PI: y = 2e⁻²ˣ
6. [A1] Writes GS. GS: y = e⁻ˣ (A cos 2x + B sin 2x) +2e⁻²ˣ
7. [B1] Uses y(0) = 5 to find A. y = 5, x = 0 ⇒ 5 = A + 2 ⇒ A = 3
8. [M1] Uses y'(0) = 1 to find B. y' = -e⁻ˣ (A cos 2x + B sin 2x) +e⁻ˣ (-2A sin 2x + 2B cos 2x) -4e⁻²ˣ
9. [A1] y' = 1, x = 0 ⇒ 1 = -3 + 2B -4 ⇒ B=4
10. [A1 CAO] Writes particular solution. ∴ y = e⁻ˣ (3 cos 2x + 4 sin 2x) +2e⁻²ˣ