= Solution
The <Debye model> replaces the crystal by a harmonic elastic continuum with acoustic dispersion $\omega=v_sk$. It counts three <polarizations> and introduces a cutoff so that the total number of vibrational modes is $3N$, where $N$ is the atom count. Periodic-box counting gives
$$
g(\omega)=\frac{V\omega^2}{2\pi^2}\left(\frac1{v_L^3}+\frac2{v_T^3}\right).
$$
For an isotropic average speed, $3/v_D^3=1/v_L^3+2/v_T^3$. With cutoff <frequency> $\omega_D$ fixed by the mode count,
$$
\boxed{g(\omega)=\frac{9N\omega^2}{\omega_D^3}\quad(0<\omega<\omega_D),\qquad
\Theta_D=\frac{\hbar\omega_D}{k_B}}.
$$
This <density of states> is quadratic because a three-dimensional shell has <volume> proportional to $k^2dk$ and the acoustic dispersion is linear. The cutoff is a mode-counting device; it is not a claim that every actual branch stays linear to the zone boundary.
Each <phonon> mode has mean thermal <energy> $\hbar\omega/(e^{\beta\hbar\omega}-1)$, using <Bose-Einstein statistics> with zero <phonon> <chemical potential>. The fixed harmonic zero-point contribution has zero <temperature> <derivative>. With $x=\hbar\omega/(k_BT)$,
$$
U_{\rm th}=9Nk_BT\left(\frac{T}{\Theta_D}\right)^3
\int_0^{\Theta_D/T}\frac{x^3}{e^x-1}dx,
$$
and differentiating at fixed <volume> gives
$$
\boxed{C_V=9Nk_B\left(\frac{T}{\Theta_D}\right)^3
\int_0^{\Theta_D/T}\frac{x^4e^x}{(e^x-1)^2}dx}.
$$
For $T\ll\Theta_D$, extend the upper limit to infinity. Expanding $1/(e^x-1)=\sum_{j\geq1}e^{-jx}$ gives $\int_0^\infty x^3/(e^x-1)dx=6\sum j^{-4}=\pi^4/15$. <integration> by parts makes the heat-capacity <integral> $4\pi^4/15$, so
$$
\boxed{C_V\simeq\frac{12\pi^4}{5}Nk_B\left(\frac T{\Theta_D}\right)^3}.
$$
Only modes with $\hbar\omega\lesssim k_BT$ are appreciably excited; their number scales as $T^3$, explaining the <power> law physically. At high <temperature> every mode contributes $k_B$, giving the Dulong-Petit limit $C_V\to3Nk_B$.
The model assumes the <harmonic approximation>, a macroscopic three-dimensional solid and acoustic continuum behaviour for the thermally populated modes. Anisotropic <sound speeds> change the low-temperature coefficient rather than its cubic <power>. Real optical branches and nonlinear acoustic dispersion limit the full-temperature approximation; electronic excitations can add a linear term in a metal, and <anharmonicity> produces <thermal expansion> and distinctions between $C_P$ and $C_V$. At <temperatures> below the lowest finite-sample mode, the bulk continuum law also fails. Thus \b[the $T^3$ result is the low-temperature lattice contribution], not a universal total <heat capacity> for every solid.
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