Let $$u(t,x) = \sum_{n=1}^3 a_n(t) \phi_n\left(\frac{x}{L}\right), \quad \text{with} \quad \phi_n\left(\frac{x}{L}\right) = \sqrt{2}\cos\left(\frac{(2n-1)\pi}{2L}x \right) \tag 1$$
which is the Galerkin approximation of the solution $u(t,x)$ of the PDE:
$$ u_{tt}(t,x) = u_{xx}(t,x)+u_{txx}(t,x) - u_t(t,x) $$
with boundary conditions $u_x(t,0) = u(t,L) = 0$.
and
$$ \ddot{a}_n(t) = -\frac{(2n-1)\pi^2}{4L^2}a_n(t) - \left( 1 + \frac{(2n-1)^2\pi^2}{4L^2}\right)\dot{a}_n(t), \quad n = 1,2,3 \tag 2 $$ with initial conditions:
$$ u(0,x) = C(x^2-x^3), \quad u_t(0,x) \equiv 0 \tag 3 $$
The objective is to determine the functions $a_n(t), \, n=1,2,3$ from the ode system and the initial conditions for $L=1$.
Attempt:
Solving each of the $3$ equations we obtain the general solutions: $$ \begin{align} a_1(t) & = c_1e^{-\frac{\pi^2}{4}t} + c_2e^{-t} \\ a_2(t) & = c_3e^{-\frac{9\pi^2}{4}t} + c_4e^{-t} \\ a_3(t) & = c_5e^{-\frac{25\pi^2}{4}t} + c_6e^{-t} \end{align} \tag 4 $$ and through the initial conditions: $$ \begin{align} a_1(0) + a_2(0) + a_3(0) & = 0 \\ \dot{a}_1(0) + \dot{a}_2(0) + \dot{a}_3(0) & = 0 \end{align} \tag 5 $$ but the above equations alone don't determine $(c_1,c_2,c_3,c_4,c_5,c_6)$ uniquely.
What other equations can be used to find these constants?
The diagonal system of ODE (2) was probably obtained as follows (I haven't checked but the $\phi_i$ seem to be orthogonal to each other with the scalar product $\int_0^L \phi_i\phi_k\mathrm{d}x$): $$\forall i=1,2,3,\quad\int_0^L\phi_i(x)\Bigl(\sum_{k=1}^{3}\ddot{a}_k(t)\phi_k(x)+\dot{a}_k(t)\phi_k(x)-a_k(t)\phi_{k,xx}(x)-\dot{a}_k(t)\phi_{k,xx}(x)\Bigr)\mathrm{d}x=0$$ The same applies to initial conditions, that is (to be computed; there might be a missing scaling factor): $$\forall i=1,2,3,\quad a_i(0)=\int_0^L\phi_i(x) C(x^2-x^3)\mathrm{d}x$$ (stemming from $\sum_{k=1}^3a_k(0)\phi_k(x)=C(x^2-x^3)$ and projection on the $\phi_i$) and $$\forall i=1,2,3,\quad \dot{a}_i(0)=\int_0^L\phi_i(x) \times0\,\mathrm{d}x=0$$ so that you can solve uniquely for your six unknown coefficients $c_i$, $i=1,\ldots,6$ as soon as $C$ is given.