= Solution
Every <eigenvector> with <eigenvalue> different from one lies in $E=\operatorname{span}\{a,b\}$, because $(\lambda-1)y=Ky\in E$. The elementary overlaps are $\langle a,a\rangle=1/2$, $\langle a,b\rangle=1/3$, $\langle b,b\rangle=1/4$. Thus
$$
La=-9a+6b,\quad Lb=-\tfrac{15}2a+5b,\quad
[L]_E=\begin{pmatrix}-9&-15/2\\6&5\end{pmatrix}.
$$
Its <characteristic polynomial> is $\lambda(\lambda+4)$, so the finite-<multiplicity> eigenpairs are
$$
\boxed{\lambda=0:\ y_0=5e^{-x}-6e^{-2x};\qquad\lambda=-4:\ y_{-4}=3e^{-x}-2e^{-2x}.}
$$
Each <eigenspace> has <dimension> one. On $E^\perp$, both integrals defining $K$ vanish, so $Ly=y$. Thus $\boxed{\lambda=1}$ is also an <eigenvalue>, of infinite <multiplicity>, and there are no others.
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