The variables x and y satisfy the differential equation dy/dx = x e^(x+y). It is given that y = 0 when x = 0.

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

1. [B1] Separate variables correctly and attempt integration of at least one side (∫e⁻ʸ dy = ∫xeˣ dx)
2. [B1] Obtain term -e⁻ʸ (B0B1 is possible)
3. [M1] Commence integration by parts and reach xeˣ ± ∫eˣ dx (B0B0M1A1 is possible)
4. [A1] Obtain xeˣ - eˣ (or equivalent)
5. [M1] Use x = 0, y = 0 to evaluate a constant, or as limits in a definite integral, in a solution with terms ae⁻ʸ, bxeˣ and ceˣ, where abc ≠ 0 (B1B1M1A1 is available if there is no constant of integration)
6. [A1] Obtain correct solution in any form (e.g. e⁻ʸ = eˣ - xeˣ)
7. [A1] Rearrange as y = -ln(1 - x) - x (or equivalent e.g. y = ln(1/(eˣ(1-x))), ISW)
8. [B1] Justify the given statement (e.g. require 1-x > 0 for the ln term to exist, hence x < 1. Must be considering the range of values of x, and must be relevant to their y involving ln(1-x))