The variables x and y satisfy the differential equation dy/dx = xey-x, and y = 0 when x = 0.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. B1 Correct separation of variables ∫e⁻ʸdy = ∫xe⁻ˣdx Condone missing integral signs.
2. B1 Obtain term -e⁻ʸ
3. *M1 Commence integration by parts and reach ±xe⁻ˣ ± ∫e⁻ˣ dx M0 if clearly using differentiation of a product.
4. A1 Complete integration and obtain -xe⁻ˣ -e⁻ˣ
5. DM1 Use x = 0 and y = 0 to evaluate a constant or as limits in a solution containing or derived from terms ae⁻ʸ, bxe⁻ˣ and ce⁻ˣ, where abc ≠ 0 Must see working for this. In a correct solution they should have -e⁻ʸ + C = -xe⁻ˣ – e⁻ˣ or equivalent. If they take logarithms before finding the constant, the constant must be of the right form.
6. A1 Correct solution in any form e.g. -e⁻ʸ = -xe⁻ˣ – e⁻ˣ A0 if constant of integration ignored or assumed to be zero.
7. A1 Obtain final answer y = -ln((x+1)e⁻ˣ) from correct working OE e.g. y = x - ln(x+1), y = ln (eˣ/(x+1)) A0 if constant of integration ignored or assumed to be zero.
8. B1 Obtain answer (y =)1-ln2 Must follow from at least 6 or7 obtained in part 6(a).