Consider a homogeneous differential equation in one dimension on *a* ≤ *x* ≤ *b*

In the study of stationary states in quantum chemistry, one would normally introduce boundary conditions, as for instance, Φ (*a*) = Φ (*b*) = 0 and solve the resulting eigenvalue problem. Solutions occur only for certain values of *E* = ∊_{n}, so-called eigenvalues, and the corresponding solutions Φ_{n} (*x*) are called eigenfunctions.

**Example 1:** Particle in a box, *i.e.*,

and for a = 0, the solutions are obtained

**Example 2:** Harmonic oscillator, *i.e.*,

with the solutions where and *H _{n}* is a

The previous two example solutions were obtained by traditional solution methods, as, for example, a series method. Instead of proceeding in this manner, we consider a general solution of the second-order differential equation (2.1):

where *u* and *v* are particular ...

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