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After elementary conversions and summing ofsimilar items, the expression takes the form:u( x, y)= a + a y + a x +0,0 0,1 1,0+ a ( y − x ) + a xy + a ( y − 3 x y)+2 2 3 20,2 1,1 0,32 1 3 4 2 2 4 3 3+ a1,2 ( xy − x ) + a0,4( y − 6 x y + x ) + a1,3( xy − x y)+35 2 3 4 4 3 2 1 5+ a0,5( y − 10x y + 5 x y) + a1,4( xy − 2 x y + x ) + ...5In general, the solution of differential equationappears in the form of the following formula:∞∑( 0, k 0, k 1, k 1, k )( , ) = ⋅ ( , ) + ⋅ ( , )u x y a Mag x y a Mag x yk = 0(3),where a0,kand a1,kthe free variables, but theMag x y andfunctions 0, k ( , )( )Mag x y are determined from the1, k,expressions:Mag0,00,1( x y)( )( )( ), = 1,Mag x, y = y,Mag x y y x2 20,2, = - ,Mag x y y y x0,33 2, = - 3 × ,( )4 2 2 4Mag0,4x, y = y - 6 y × x + x ,- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -k / 2 m 2mk −2m( −1)× x × yMag0,k ( x,y)= k!∑m=0 (2m)!(k − 2m)!(4).It is possible to write down the recursion formula:k / 22mk −2mMag0 , k ( x,y)= ∑ amx × ym=0( k − 2m+ 2)( k − 2m+ 1)a0= 1;am= −am−12m(2m− 1)(5).Or even in the economic form (from the point ofview of a quantity of performed operations):k / 2Mag 0 , k ( x,y)= ∑ am=0m (6)ak0 = y ; am= −am−1( k − 2m+ 2)( k − 2m+ 1) x×2m(2m− 1) yThe second part of the functions entering formula(3) takes the form:1,01,1( )( )Mag x, y = x,Mag x, y = y×x,1Mag1,2( x y)= y × x-x3Mag x y = y × x- y×x1,32 3, ,( )3 3, ,1Mag1,4( x, y)= y × x- 2y × x + x54 2 3 5- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -k / 2 m 2m+1 k −2m( −1)× x × yMag1,k ( x,y)= k!∑m=0 (2m+ 1)!( k − 2m)!(7).In this case, recursion formula takes the form:Maga1 , k ( x,y)=k / 2∑ a x × ym=01;am= −am10 = −Maga(8).m2m+1k −2m( k − 2m+ 2)( k − 2m+ 1)(2m+ 1)2mOn the other hand, in the economic form:1 , k ( x,y)=k / 2∑ am=0mk0 = xy ; am= −am−1(9).( k − 2m+ 2)( k − 2m+ 1) x×(2m+ 1)2myThese are the well-studied harmonic polynomials.The only thing what the author did not find in thereference of mathematical literature is this form ofrecord and such recurrent expressions. The informationon the harmonic polynomials can be found at (Люк1980).Further, let us examine equation (1) in the morecommon form:2222Annual Proceedings of Vidzeme University College “ICTE in Regional Development”, 2006132