  An algorithm for providing analytical solutions to Schr\"{o}dinger's equation
with non-exactly solvable potentials is elaborated. It represents a symbiosis
between the logarithmic expansion method and the techniques of the
superymmetric quantum mechanics as extended toward non shape invariant
potentials. The complete solution to a given Hamiltonian $H_{0}$ is obtained
from the nodeless states of the Hamiltonian $H_{0}$ and of a set of
supersymmetric partners $H_{1}, H_{2},..., H_{r}$. The nodeless states (dubbed
"edge" states) are unique and in general can be ground or excited states. They
are solved using the logarithmic expansion which yields an infinite systems of
coupled first order hierarchical differential equations, converted later into
algebraic equations with recurrence relations which can be solved order by
order. We formulate the aforementioned scheme, termed to as "Supersymmetric
Expansion Algorithm'' step by step and apply it to obtain for the first time
the complete analytical solutions of the three dimensional Hulth\'en--, and the
one-dimensional anharmonic oscillator potentials.
