Answer only one of the following two alternatives. EITHER Show that the substitution v = 1/y reduces the differential equation (2/y³) (dy/dx)² + (1/y²) (d²y/dx²) + (5/y) = 17 + 6x – 5x² to the differential equation d²v/dx² + 2dv/dx + 5v = 17 + 6x – 5x². Hence find y in terms of x, given that when x = 0, y = 1/2 and dy/dx = -1.

Mark scheme

Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.

1. [B1] dv/dx = -1/y² dy/dx
2. [M1A1] d²v/dx² = -1/y² d²y/dx² + (2/y³) (dy/dx)²
3. [A1] (2/y³) (dy/dx)² + (1/y²) (d²y/dx²) + (5/y) = d²v/dx² + 5v = 17 + 6x – 5x² (AG)
4. [M1A1] m² + 2m + 5 = 0 ⇒ m = -1 ± 2i ⇒ CF : e⁻ˣ (A cos 2x + B sin 2x)
5. [M1] PI : v = px² + qx + r ⇒ v' = 2px + q ⇒ v'' = 2p
6. [M1] Equate coefficients: 5p = -5 4p + 5q = 6 2p + 2q + 5r = 17
7. [A1] p = -1, q = 2, r = 3
8. [A1] GS : v = e⁻ˣ (A cos 2x + B sin 2x) + 3 + 2x – x²
9. [B1] When x = 0 y = 1/2 ⇒ v = 2 and dy/dx = -1 ⇒ dv/dx = 4
10. [B1] 2 = A + 3 ⇒ A = -1
11. [M1] v' = e⁻ˣ (-2A sin 2x + 2B cos 2x) - e⁻ˣ (A cos 2x + B sin 2x)
12. [A1] 4 = 2B + 1 + 2 ⇒ 2B = 1 ⇒ B = 1/2
13. [A1] ⇒ y = [e⁻ˣ (1/2 sin 2x - cos 2x) + 3 + 2x - x²]⁻¹