Answer only one of the following two alternatives. OR Given that 1/y d²y/dx² - 6/y² (dy/dx)² + 2/y³ (dy/dx) + 3y = 25e^{-2x} and that v = y³, show that d²v/dx² - 6dv/dx + 9v = 75e^{-2x}. Find the particular solution for y in terms of x, given that when x = 0, y = 2 and dy/dx = 1.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.

1. [B1B1] (a) Uses substitution to obtain v-x equation. v = y³ ⇒ v' = 3y²(dy/dx) v'' = 6y(dy/dx)² + 3y²(d²y/dx²)
2. [M1] 1/3 d²v/dx² - 2 dv/dx + 3v = 25e^{-2x}
3. [A1] ⇒ d²v/dx² - 6dv/dx + 9v = 75e^{-2x} (AG)
4. [M1] (b) Finds CF. m² – 6m + 9 = 0 ⇒ m = 3
5. [A1] v = Ae³ˣ + Bxe³ˣ
6. [M1] Finds PI. v = ke⁻²ˣ ⇒ v' = −2ke⁻²ˣ ⇒ v'' = 4ke⁻²ˣ
7. [A1] 4k + 12k + 9k = 75k ⇒ k = 3
8. [A1] v = Ae³ˣ + Bxe³ˣ + 3e⁻²ˣ
9. [B1] Uses initial conditions to find constants. x = 0, y = 2, v = 8 ⇒ 8 = A + 3 ⇒ A = 5
10. [M1A1] v' = 15e³ˣ + 3Bxe³ˣ + Be³ˣ − 6e⁻²ˣ
11. [A1] x = 0, y = 2, y' = 1 ⇒ v' = 12
12. [A1] 12 = 15 + B − 6 ⇒ B = 3
13. [A1] Writes solution of y-x equation explicitly. y³ = v = 5e³ˣ + 3xe³ˣ + 3e⁻²ˣ
14. [A1] y = (5e³ˣ + 3xe³ˣ + 3e⁻²ˣ)^(1/3)