The value, V, of a product, t years after it was launched, is modelled by the differential equation dV/dt = kV/(t + 1), where k is a constant. Given that V = V₀ when t = 0, (a) show that V = V₀(t + 1)^k.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. M1 for separating the variables correctly: ∫(1/V) dV = ∫(k/(t + 1)) dt.
2. A1 for integrating both sides correctly: ln|V| = k ln|t + 1| + C.
3. M1 for using the initial condition (t=0, V=V₀) to find C: ln(V₀) = k ln(1) + C => C = ln(V₀).
4. A1 for correctly rearranging to V = V₀(t + 1)^k, showing sufficient steps. (e.g., ln(V) - ln(V₀) = k ln(t+1) => ln(V/V₀) = ln((t+1)^k) => V/V₀ = (t+1)^k).