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Now more general equation of the form will beexamined:2 2 2∂ u ∂ u ∂ u+ b + d + wu = 0∂x ∂x ∂x2 2 21 2 3where b, d,w the arbitrary constants. Therecurrent expressions for this equation will be thefollowing:Magm / 2n/ 22i+2 j m−2in−2j20 , m,n = ∑ ∑ ai,j x1x2x3Ge(i + j,wx1)i= 0 j=0a 0 ,0 = 1;( m − 2i+ 2)( m − 2i+ 1)ai,j = −ai−1,jb − ai,j−(2i+ 2 j)(2i+ 2 j − 1)1( n − 2 j + 2)( n − 2 j + 1)d(2i+ 2 j)(2i+ 2 j − 1)Magm / 2n/ 22i+2 j+1 m−2in−2j21 , m,n = ∑ ∑ ai,j x1x2x3Ge(i + j + 1, wx1)i= 0 j=0a 0 ,0 = 1;( m − 2i+ 2)( m − 2i+ 1)ai,j = −ai−1,jb − ai,j−(2i+ 2 j)(2i+ 2 j + 1)1( n − 2 j + 2)( n − 2 j + 1)d(2i+ 2 j)(2i+ 2 j + 1)It is possible to read about the particular andfundamental solutions of equation (23) in (Полянин2001). But there is no similar solutions among theseparticular ones to those given in the article, as there aregiven only the solutions obtained by the method ofseparation of variables, and in these formulas generalfunctions of Herman are used which are obtained bythe method of dual substitution.FOUR-DIMENSIONAL EQUATION OFLAPLACE AND DUAL SUBSTITUTION2 2 2 2∂ u ∂ u ∂ u ∂ u+ + + = 0∂x ∂x ∂x ∂x2 2 2 21 2 3 4(27)The general solution of this differential equation is:∞ k k −m1( 1, 2, 3,4 ) = ∑ ∑ ∑ ( 0, m )1, m2 , m3 0, m1 , m2 , m+3 1. m1 , m2 , m3 1, m1 , m2 , m3(28),u x x x x a Mag a Magawhere 0, m , m , mcoefficients,1 2 3k = 0 m = 0 m = 0and1 2m3 = k −m1 −m2a1, m , m , m1 2 3the freethe functions Mag0, m ( )1, m2 , mx3 1, x2, x3,x4and Mag ( x , x , x , x ) are1, m1 , m2 , m31 2 3 4determined from the following formulas:Mag0, m , m , m1 2 3= mm1 m2m3p2 2 22 p m1 −2i m2−2j m3−2l( −1 ) p!x1 x2 x3 x4F∑∑∑ (29)i= 0 j= 0 l=0 i! j! l! ( 2 p) !( m1 − 2 i) !( m2 − 2 j) !( m3− 2 l)!Annual Proceedings of Vidzeme University College “ICTE in Regional Development”, 2006144