= Solution
The unit integrals of the profiles make $N_i(t)=\int_0^\infty n_i(\tau,t)\,d\tau$ the phase populations, and $f_i$ their normalized <cell-cycle age densities>. Assume the age tails vanish and the following integrals are finite. For a fixed profile define its mean exit <hazard function> by
$$
\boxed{\Lambda_i=\int_0^\infty\Gamma_i(\tau)f_i(\tau)\,d\tau,\qquad i=1,2.}
$$
These constants have units of inverse time. Integrating the <age-structured population equations> and using the <boundary conditions> gives
$$
\begin{aligned}
N_1'&=-[n_1(\tau,t)]_0^\infty-\int_0^\infty\Gamma_1n_1\,d\tau
=n_1(0,t)-\Lambda_1N_1,\\
N_2'&=-[n_2(\tau,t)]_0^\infty-\int_0^\infty\Gamma_2n_2\,d\tau
=n_2(0,t)-\Lambda_2N_2.
\end{aligned}
$$
The incoming <fluxes> are $n_1(0,t)=2\Lambda_2N_2$ and $n_2(0,t)=\Lambda_1N_1$, so
$$
\boxed{\frac{d}{dt}\begin{pmatrix}N_1\\N_2\end{pmatrix}
=\begin{pmatrix}-\Lambda_1&2\Lambda_2\\\Lambda_1&-\Lambda_2\end{pmatrix}
\begin{pmatrix}N_1\\N_2\end{pmatrix}.}
$$
The <separable-profile closure for an age-structured population> has a qualification: arbitrary age-dependent <hazard functions> do not give this constant-coefficient <ordinary differential equation> for arbitrary <initial conditions>. Without fixed $f_i$, the effective exit rates are $\int\Gamma_i n_i\,d\tau/N_i$ and can vary with time. Moreover, an exact positive <separation of variables> with both profiles fixed forces $N_2/N_1$ to be constant through the incoming <fluxes>, hence selects a common exponential mode. The displayed <ordinary differential equation> is the necessary integrated system for such profiles; it is also exact for arbitrary populations when both <hazard functions> are age independent.
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