I. $\lambda > 0 .$

The general solution for (1.2) is given by

$$G(x) = C_1 e^{\sqrt{\lambda} x} + C_2 e^{- \sqrt{\lambda} x}$$

We need to satisfy the boundary conditions

$$G(0) = 0,\;\; G(L) = 0$$

with the choice of $C_1,\;\;C_2$ and $\lambda.$

$$\begin{eqnarray*} C_1 + C_2 &=& 0,\;\; \mbox{ follows from } G(0) = 0\\ C_1 e^{... ...^{- \sqrt{\lambda} L} &=& 0 ,\;\; \mbox{ follows from } G(L) = 0 \end{eqnarray*}$$

For $\lambda > 0$ this system has the only solution

$$C_1 =0,\;\; C_2 =0.$$

Hence, the condition $\lambda > 0$ does not bring us any nontrivial solutions.