Issue 49

V. Matveenko et alii, Frattura ed Integrità Strutturale, 49 (2019) 225-242; DOI: 10.3221/IGF-ESIS.49.23

ì ï ï ï ï ï ï î

ü ï ï ï ý ï ï ï þ

( ) P D x x æ ù + í (4) (4) ln m m

ö+ ÷

k

1

¥ å

2 ç - ÷ ç ÷ çè ø m

é ê ë

(4) v x

=

,

( )

ú û

k

=

m

0

(1)

(2)

(3)

(4)

(3)

(4 )

where the coefficients , m m P P P P D D can be determined and calculated E-resource (www. , , , , m m m m icmm.ru/compcoeff). Then, solving Eqns. (39) as a homogeneous one, we find two more partial solutions (5) (6) , k k 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 v v . The form of this differential equation indicates that the point

   

   

  

1 km  

   

   

 

  

1 km  

 





 

  P D x x  (6) (6) ln

(6) v x k

 

2







(5) v x k

(5) P x m

2



,





(43)

m

m



m

0



m

0

(5)

(6)

(6)

, , m m m P P D can be determined and calculated E-resource (www. icmm.ru/compcoeff).

where the coefficients

(1) k w w w w (2) (3) (4) , , , k k k

, (1) , , k k k k k k v v v v v v (2) (3) (4) (5) (6) , , ,

At the next stage, we use the partial solutions relation (40) to derive six partial solutions (1) ,

and the obtained

(2)

(3)

(4)

(5)

(6)

, k u u u u u u , which are expressed as , , , k k k k k

 2 1 1  2 1 1  



  

   

  

1 km  

x x 

 

(1)

(1) E x m

2



u





    kx x S 



 2 2   

k



m

0



  

   

  

1 km  

x x 

 

(2)

(2) E x m

2



u





    kx x S 



 2 2   

k



m

0

 x x 



           

       

  

1 km  

2 1

 





  E G x x  (3) (3) ln

(3)

2



u





(44)

    kx x S   1



 2 2   

k

m

m



m

0

 x x 



  

1 km  

2 1

 





  E G x x  (4) (4) ln

(4)

2



u





    kx x S   1



 2 2   

k

m

m



m

0

 1 x x 



   

  

1 km  

 

(5)

(5) E x m

2



u





    kx x S   1



 2 2   

k



m

0

 1 x x 



   

  

1 km  

 





  E G x x  (6) (6) ln

(6)

2



u





    kx x S   1



 2 2   

k

m

m



m

0

(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

where the coefficients

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)

    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                                 1 2 3 4 5 6 (1) (1) k (2) (2) k (3) (3) k (4) (4) k (5) (5) k (6) (6) k 1 1 2 2 3 3 4 4 5 5 6 6 (1 1 k k k k k k k k k k k k k k k k

(45)

235