The variables x and t satisfy the differential equation dx/dt = x²(1 + 2x), and x = 1 when t = 0. Using partial fractions, solve the differential equation, obtaining an expression for t in terms of x.

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

1. [B1] State a suitable form of partial fractions for 1 / (x²(1+2x))
2. [M1] Use a relevant method to determine a constant
3. [A1] Obtain one of A = − 2, B = 1 and C = 4
4. [A1] Obtain a second value
5. [A1] Obtain the third value
6. [M1] Separate variables correctly and integrate at least one term
7. [B3 FT] Obtain terms -2 ln x - (1/x) + 2 ln(1+2x) and t
8. [M1] Evaluate a constant, or use limits x = 1, t = 0 in a solution containing terms t, a ln x and b ln(1+2x) , where ab ≠ 0
9. [A1] Obtain a correct expression for t in any form, e.g. t = -(1/x) + 2 ln((1+2x)/x) + 1