Applying Lienard–Schipar's method to solving of homogeneous fractional differential Euler-type equations on an interval

We present the solution of the homogeneous fractional differential Euler-type equation on the half-axis in the class of functions representable by the fractional integral of order $\alpha$ with the density of $L_1(0;1)$. Using the method of Hermitian forms (Lienard–Schipar's method), solvability conditions are obtained for the cases of two, three and a finite number of derivatives. It is shown that in the case when the characteristic equation has multiple roots original equation admits a solution with logarithmic singularities.