12. 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 $$\frac { d P } { d t } - \frac { P } { t \left( 1 + t ^ { 2 } \right) } = \frac { t e ^ { - t } } { \sqrt { 1 + t ^ { 2 } } }$$ Find \(P\) as a function of \(t\). [You may assume that \(\lim _ { t \rightarrow \infty } t e ^ { - t } = 0\) ].
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