= Solution
After the short prompt adjustment, the <prompt jump approximation> takes the <neutron> amplitude to remain nearly in balance with the slowly changing <delayed neutron precursor> reservoir. For the source-free model,
$$
0\simeq\frac{\rho-\beta}{\Lambda}n+\lambda c,
\qquad n\simeq\frac{\Lambda\lambda}{\beta-\rho}c.
$$
Substitute this quasi-steady relation into the precursor equation:
$$
\frac{dc}{dt}=\frac{\beta\lambda}{\beta-\rho}c-\lambda c
=\frac{\rho\lambda}{\beta-\rho}c.
$$
For constant coefficients,
$$
\boxed{c(t)=c_0\exp\!\left(\frac{\rho\lambda}{\beta-\rho}t\right).}
$$
The approximation is meaningful in the stated delayed-<neutron> regime with $\rho<\beta$ and a separated fast transient. The <delayed neutron precursor> reservoir remains continuous across an idealized instantaneous <reactivity> step; the approximation does not resolve the short prompt adjustment itself.
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