A-LevelFurther MathematicsDifferential equations9231/12 · May/June 2018 · Q710 Marks
Find the particular solution of the differential equation 49(d²y/dx²) + 14(dy/dx) + y = 49x + 735, given that when x = 0, y = 0 and dy/dx = 0.
- Find the particular solution of the differential equation 49(d²y/dx²) + 14(dy/dx) + y = 49x + 735, given that when x = 0, y = 0 and dy/dx = 0.[10]
Mark scheme
Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.
- M1 The auxiliary equation is 49n² + 14n + 1 = 0 ⇒ (7n+1)² = 0 ⇒ n = -1/7
- A1 CF: y = (A+Bx)e^(-x/7)
- M1 PI: y = px+q, y'=p and y''=0. Correct form of PI and derivatives.
- M1 Substitutes in equation correctly.
- A1 14p + px + q = 49x + 735 ⇒ p=49, q=49.
- A1 FT Thus y = (A+Bx)e^(-x/7) + 49(x+1).
- B1 y=0 when x=0 gives A+49 = 0 ⇒ A = -49.
- M1 Differentiating their y. y' = -1/7(A+Bx)e^(-x/7) + Be^(-x/7) + 49
- A1 y'=0 when x=0 gives -1/7(A) + B + 49 = 0 ⇒ B = -56.
- A1 Thus y = -(49+56x)e^(-x/7) + 49(x+1).
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