Issue 49

V. Matveenko et alii, Frattura ed Integrità Strutturale, 49 (2019) 225-242; DOI: 10.3221/IGF-ESIS.49.23 235 ( ) ( ) 1 (4) (4) (4) 2 0 ln , k m m m k m v x P D x x æ ö+ ÷ ¥ ç - ÷ ç ÷ çè ø = ì ü ï ï ï ï ï ï é ù = + í ý ê ú ë û ï ï ï ï ï ï î þ å where the coefficients (1) (2) (3) (4) (3) (4 ) , , , , , m m m m m m P P P P D D can be determined and calculated E-resource (www. icmm.ru/compcoeff) . Then, solving Eqns. (39) as a homogeneous one, we find two more partial solutions (5) (6) , k k v v . The form of this differential equation indicates that the point 0 x  is a regular singular point. The construction of partial solutions in the form of the generalized power series is accomplished in the framework of the proposed approach. The obtained partial solutions are expressed as   1 (5) (5) 2 0 , km k m m v x P x                          1 (6) (6) (6) 2 0 ln km k m m m v x P D x x                     (43) where the coefficients (5) (6) (6) , , m m m P P D can be determined and calculated E-resource (www. icmm.ru/compcoeff ). At the next stage, we use the partial solutions (1) (2) (3) (4) , , , k k k k w w w w , (1) (2) (3) (4) (5) (6) , , , , , k k k k k k v v v v v v and the obtained relation (40) to derive six partial solutions (1) (2) (3) (4) (5) (6) , , , , , k k k k k k u u u u u u , which are expressed as         1 (1) (1) 2 0 2 1 1 2 2 km k m m x x u E x kx x S                                 1 (2) (2) 2 0 2 1 1 2 2 km k m m x x u E x kx x S                                     1 (3) (3) (3) 2 0 2 1 ln 1 2 2 km k m m m x x u E G x x kx x S                          (44)             1 (4) (4) (4) 2 0 2 1 ln 1 2 2 km k m m m x x u E G x x kx x S                                  1 (5) (5) 2 0 1 1 2 2 km k m m x x u E x kx x S                                     1 (6) (6) (6) 2 0 1 ln 1 2 2 km k m m m x x u E G x x kx x S                          where the coefficients (1) (2) (3) (4) (5) (6) (3) (4) (6) , , , , , , , , m m m m m m m m m E E E E E E G G G for any value of 0 m  can be determined and calculated E-resource (www. icmm.ru/compcoeff ). The general solutions for , , k k k u v w are given by                                             (1) (2) (3) (4) (5) (6) 1 2 3 4 5 6 (1) (2) (3) (4) (5) (6) 1 2 3 4 5 6 (1) (2) (3) (4) (5) (6) 1 2 3 4 5 6 (1 1 k k k k k k k k k k k k k k k k k k k k k k k u x C u x C u x C u x C u x C u x C u x v x C v x C v x C v x C v x C v x C v x v x C v x C v x C v x C v x C v x C v x w x C w                                               ) (2) (3) (4) 2 3 4 k k k x C w x C w x C w x       (45)