= Solution
Set $\lambda'=0$. A decaying <normal mode> $e^{-kt}$ obeys
$$
C_sC_d k^2-[(\lambda+\gamma)C_d+\gamma C_s]k+\lambda\gamma=0.
$$
For $C_d\gg C_s$, its sum and product of roots give
$$
\boxed{k_1\simeq\frac{\lambda+\gamma}{C_s},\quad k_2\simeq\frac{\lambda\gamma}{C_d(\lambda+\gamma)}.}
$$
Initially $T_s=T_d=T_0$, so $a_1+a_2=1$ and $a_1k_1+a_2k_2=\lambda/C_s$. To leading order,
$$
\boxed{a_1\simeq\frac{\lambda}{\lambda+\gamma},\qquad a_2\simeq\frac{\gamma}{\lambda+\gamma}.}
$$
On the slow <normal mode>, the first equation gives $T_d/T_s=(\lambda+\gamma-C_sk_2)/\gamma\simeq(\lambda+\gamma)/\gamma$, verifying proportionality after the fast transient has decayed. The small parameter must also make $k_2/k_1$ small; the supplied positive coefficients and $C_d\gg C_s$ satisfy this.
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