On Approximate Euler Differential Equations

and Applied Analysis 3 is called the auxiliary equation of the Euler differential equation 2.1 , and every solution of 2.1 is of the form yh x ⎧ ⎪⎨ ⎪⎩ c1x m1 c2x2 if m1 and m2 are distinct roots of 2.2 , c1 c2 lnx x 1−α /2 if 1 − α 2 is a double root of 2.2 , 2.3 where c1 and c2 are complex constants see 21, Section 2.7 . Theorem 2.1. Let α and β be complex constants such that no root of the auxiliary equation 2.2 is a nonnegative integer. If the radius of convergence of power series ∑∞ m 0 amx m is at least ρ > 0, then every solution y : 0, ρ → C of the inhomogeneous Euler differential equation 1.4 can be expressed by