Challenging +1.8 This is a first-order linear ODE requiring identification of integrating factor μ = exp(∫-1/(t(1+t²))dt), which involves partial fractions to get ln(t/√(1+t²)). Then solving μP' = μ·(te^(-t)/√(1+t²)) requires integration by parts and careful application of two boundary conditions (P(0)=0 and lim P = 5). The multi-step nature, non-trivial integrating factor, and boundary condition handling make this substantially harder than routine DE questions but still within standard Further Maths scope.
The population density \(P\), in suitable units, of a certain bacterium at time \(t\) hours is to be modelled by a differential equation. Initially, the population density is zero, and its long-term value is 5.
The model uses the differential equation
$$\frac{dP}{dt} - \frac{P}{t(1 + t^2)} = \frac{te^{-t}}{\sqrt{1 + t^2}}$$
Find \(P\) as a function of \(t\). [You may assume that \(\lim_{t \to \infty} te^{-t} = 0\)]. [11]
The population density $P$, in suitable units, of a certain bacterium at time $t$ hours is to be modelled by a differential equation. Initially, the population density is zero, and its long-term value is 5.
The model uses the differential equation
$$\frac{dP}{dt} - \frac{P}{t(1 + t^2)} = \frac{te^{-t}}{\sqrt{1 + t^2}}$$
Find $P$ as a function of $t$. [You may assume that $\lim_{t \to \infty} te^{-t} = 0$]. [11]
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q12 [11]}}