The variables x and y satisfy the differential equation dy/dx = ky³e⁻ˣ, where k is a constant. It is given that y = 1 when x = 0, and that y = √e when x = 1. Solve the differential equation, obtaining an expression for y in terms of x.

Mark scheme

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

1. [B1] Separate variables correctly and attempt integration of at least one side
2. [B1] Obtain term -1/(2y²), or equivalent
3. [B1] Obtain term -ke⁻ˣ
4. [M1] Use a pair of limits, e.g. x = 0, y = 1 to obtain an equation in k and an arbitrary constant c
5. [M1] Use a second pair of limits, e.g. x = 1, y = √e, to obtain a second equation and solve for k or for c
6. [A1] Obtain k = 1/2 and c = 0
7. [A1] Obtain final answer y = e^(x/2), or equivalent