Let $X$ be a real analytic Riemannian manifold with a boundary $\partial X$. Denote its interior by $\mathop {X}\limits ^{\circ }$ and its metric by $g$. Introduce on $X$ a conformal metric $h$ defined by $h = p^{-2}g$ where $p(x)$ is a real analytic on $X$ such that $p(x) > 0$ in $\mathop {X}\limits ^{\circ }$, $p(x) = 0$ and $dp \neq 0$ on $\partial X$. Under the metric $h$, $X$ becomes a complete non-compact Riemannian manifold with a corresponding Laplacian $\Gamma _h$. Consider solutions of the differential equation. $(*)\quad \Gamma _hu + \lambda q(x)u = 0 \hbox { on }X$ where $q(x)$ is a real analytic function on $\partial X$ and $\lambda \in \mathbb {C}$. Our main result is a representation theorem for all solutions of equation (*). The theorem is a generalization of a representation formula established by Helgason and Minemura for solutions of the Helmholtz equation on hyperbolic space.