$C = p^{2}$

When the constant is positive, the separable equations are

$$X^{\prime\prime} - p^{2} X = 0,$$ (2.26)

$$\ddot{T} - c^{2}p^{2} T = 0.$$ (2.27)

Equations (2.26) and (2.27) have constant coefficients and so can be solved in the usual way. For example, for (2.26), we set

$$X(x) = e^{\lambda x}, \qquad \Rightarrow \qquad X^{\prime\prime}(x) = \lambda^{2}e^{\lambda x} = \lambda^{2}X(x).$$

Substituting into (2.26) we get

$$\lambda^{2}X - p^{2}X = 0, \qquad \Rightarrow \qquad \lambda^{2} = p^{2}.$$

Thus,

$$\lambda = \pm p,$$

and the solution to (2.26) is

$$X(x) = A e^{p x} + B e^{-p x}.$$ (2.28)

To satisfy the boundary conditions in $x$ we find that the only solution is the trivial solution with

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