where ak(x) ∈ C∞[0, 1] (k = 1, . . . , n− 1) and f(x) ∈ C∞[0, 1] are given functions, φ(x) ∈ C∞[0, 1] is the unknown function, and n is a positive integer. Here C∞[0, 1] is the class of infinitely differentiable functions on the interval [0, 1]. Equation (1) with f ≡ 0 is said to be homogeneous. If aj(x) ≡ const (j = 0, . . . , n − 1), then this equation coincides with the Euler equation. Since the coefficient of φ(x) in Eq. (1) vanishes for x = 0, it follows that this equation has a singularity at this point. The singularity order of Eq. (1) is equal to n. Differential equations of the form (1) arise in numerous problems of mathematical physics. Such equations include Bessel, Legendre, Chebyshev–Laguerre, and other equations [1]. Some special cases of homogeneous equations of the form (1) have been rather comprehensively studied in the class of analytic functions of the real variable x [1–3] as well as in the class of twice continuously differentiable functions on the interval [0, 1] (see [4, 5]). In [1–5], it was assumed that the coefficients of Eq. (1) are analytic with respect to the real variable x in a neighborhood of zero. The aim of the present paper is to obtain necessary and sufficient conditions for the unique solvability of Eq. (1) in the class C∞[0, 1] and indicate a solution method. By Pn(λ) we denote the polynomial