The variables x and y satisfy the differential equation dy/dx = (y - 1) / ((x + 1)(x + 3)). It is given that y = 2 when x = 0. Solve the differential equation, obtaining an expression for y in terms of x.

Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

1. [B1] Separate variables correctly and integrate at least one side
2. [B1] Obtain term ln(y – 1)
3. [M1] Carry out a relevant method to determine A and B such that 1/((x+1)(x+3)) = A/(x+1) + B/(x+3)
4. [A1] Obtain A = 1/2 and B = –1/2
5. [A1 FT + A1 FT] Integrate and obtain terms (1/2)ln(x+1) – (1/2)ln(x+3), or equivalent (FT is on A and B)
6. [M1] Use x = 0, y = 2 to evaluate a constant, or as limits in a solution containing terms of the form a ln(y – 1), b ln(x + 1) and c ln(x + 3), where abc ≠ 0
7. [A1] Obtain correct answer in any form
8. [A1] Obtain final answer y = 1 + √((x+1)/(x+3)), or equivalent