Issue 49

V. Matveenko et alii, Frattura ed Integrità Strutturale, 49 (2019) 225-242; DOI: 10.3221/IGF-ESIS.49.23 234         2 2 2 2 2 1 1 4 18 1 13 2 f x x x x x k                         3 4 3 4 3 4 6 1 2 1 1 f x x x x f x x x      (39)                                     2 2 2 0 2 2 0 2 2 1 2 3 2 3 2 3 1 1 1 1 ; 1 4 2 1 1 4 1 1 1 2 1 1 4 4 1 1 2 1 1 5 2 1 2 x x x k x x x x x x x x k k x x x x x k k x x x x x x x k k                                                Moreover, the results of these transformations is a set of equations, which relate the function k u to the functions , k k w v and their derivatives ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 1 4 1 4 2 1 2 1 2 2 k k k x x d w x dw x u x x x x x x x k S S dx dx a ì - ïï = - + - - íï - + + ïî ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 4 1 1 1 1 2 1 2 1 2 k k k dv x x x k S w x kSx x k x S v x dx a a ü é ùïï é ù - - + + + + - - + - + ê ú ý ê ú ë û ï ê ú ë ûïþ (40) Eqn. (38) is independent of (39) and represents a linear differential equation of the fourth order with respect to the function k w . Eqn. (39) can be considered as a differential equation of the second order with respect to the function k v with right –hand side depending on k w . Such a peculiarity of the differential Eqns. (28) and the obtained relation (30) allows us to determine the sequence of constructing partial solutions for functions , , k k k w v u . The hang of this sequence can be outlines as follows. First, from the solution of Eqn. (38) we obtain four partiсular solutions (1) (2) (3) (4) , , , k k k k w w w w , written as                 1 1 (1) (1) (2) (2) 2 2 0 0 1 1 (3) (3) (3) (4) (4) (4) 2 2 0 0 , , ln , ln k k m m k m k m m m k k m m k m m k m m m m w x A x w x A x w x A B x x w x A B x x                                                                                   (41) where the coefficients (1) (2) (3) (4) (3) (4) , , , , , m m m m m m A A A A B B can be determined and calculated E-resource (www. icmm.ru/compcoeff) . Then substituting the obtained particular solutions (1) (2) (3) (4) , , , k k k k w w w w into the right-hand side of Eqn. (28. b) and solving it as an inhomogeneous equation we arrive at four partial solutions (1) (2) (3) (4) , , , k k k k v v v v , which can be written as ( ) ( ) 1 1 (1) (2) (1) (2) 2 2 0 0 , , k k m m m m k k m m v x P x v x P x æ ö æ ö + - ÷ ÷ ¥ ¥ ç ç + + ÷ ÷ ç ç ÷ ÷ ç ç è ø è ø = = é ù é ù ê ú ê ú = = ê ú ê ú ê ú ê ú ë û ë û å å ( ) ( ) 1 (3) (3) (3) 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 (42)