$C=0$

In this case, the equations (2.24) and (2.25) become

$$X^{\prime\prime} = 0, \qquad \Rightarrow \qquad X(x) = A x + B,$$

and

$$\ddot{T} = 0, \qquad \Rightarrow \qquad T(t) = D t + E,$$

for constants $A$, $B$, $D$ and $E$. However, the boundary conditions (2.17) imply that

$$A = 0 \qquad \hbox{and} \qquad B = 0.$$

Thus, the only solution, with $C=0$, it is trivial solution.