A-LevelFurther MathematicsDifferential equations9231/12 · May/June 2016 · Q911 Marks
Find the value of the constant k such that y = kx²e²ˣ is a particular integral of the differential equation d²y/dx² - 4 dy/dx + 4y = 4e²ˣ. (*) Hence find the general solution of (*). Find the particular solution of (*) such that y = 3 and dy/dx = -2 when x = 0.
- Find the value of the constant k such that y = kx²e²ˣ is a particular integral of the differential equation d²y/dx² - 4 dy/dx + 4y = 4e²ˣ. (*)[4]
- Hence find the general solution of (*).[3]
- Find the particular solution of (*) such that y = 3 and dy/dx = -2 when x = 0.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.
- [B1] y' = 2kxe²ˣ + 2kx²e²ˣ
- [B1] y'' = 2ke²ˣ + 8kxe²ˣ + 4kx²e²ˣ
- [M1] 2ke²ˣ + 8kxe²ˣ + 4kx²e²ˣ - 8kxe²ˣ – 8kx²e²ˣ + 4kx²e²ˣ = 4e²ˣ
- [A1] ⇒2k=4k=2
- [M1] m²-4m+4=0 ⇒ (m-2)² = 0 ⇒ m = 2
- [A1] CF: y = Ae²ˣ + Bxe²ˣ
- [A1] y = Ae²ˣ + Bxe²ˣ + 2x²e²ˣ
- [B1] y = 3 when x = 0 ⇒ A = 3
- [M1] y' = 2Ae²ˣ + B(e²ˣ + 2xe²ˣ)+(4xe²ˣ + 4x²e²ˣ)
- [A1] y' = -2 when x = 0 and A = 3 ⇒ B = -8
- [A1] ⇒ y=3e²ˣ - 8xe²ˣ + 2x²e²ˣ
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