= Solution
Substitution of the separated exponential profile into the <age-structured population equation> gives $r'=-(\gamma+\mu(a))r$. Therefore
$$
r(a)=C\exp\!\left[-\gamma a-\int_0^a\mu(s)\,ds\right].
$$
For $C\ne0$, the birth boundary condition cancels $C$ and becomes the <Euler-Lotka equation>
$$
\boxed{\int_0^\infty b(a)\exp\!\left[-\gamma a-\int_0^a\mu(s)\,ds\right]da=1}.
$$
The survival probability to age $a$ is $e^{-\mu a}$ for constant death rate. Hence the expected lifetime offspring count, or <net reproduction rate>, is
$$
\boxed{R_0=\int_0^\infty Ba^pe^{-(\lambda+\mu)a}\,da
=\frac{Bp!}{(\lambda+\mu)^{p+1}}}.
$$
The <Gamma integral> evaluates the <Euler-Lotka equation> as $Bp!/(\lambda+\mu+\gamma)^{p+1}=1$, requiring $\lambda+\mu+\gamma>0$. Its real solution and profile are
$$
\boxed{\gamma=(Bp!)^{1/(p+1)}-\lambda-\mu,\qquad
n(a,t)=C e^{\gamma t}e^{-[(Bp!)^{1/(p+1)}-\lambda]a}}.
$$
Thus \b[$B^*=(\lambda+\mu)^{p+1}/p!$], and $B>B^*$ gives positive growth. The profile is a nonzero formal separated solution for all positive $B$; a finite total population additionally requires $\gamma+\mu>0$, equivalently $(Bp!)^{1/(p+1)}>\lambda$. A decaying-in-time mode need not have an integrable age profile.
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