Using the assumption G(q, r, s) = e ^{∫ y(q)dq + ∫x(r)dr + ∫ z(s)ds} for solving some kinds of linear third order P.D.Es. with homogenous terms

Abstract

The preset paper aims at finding the complete solution of certain types of linear partial differential equations of third order with constant coefficients that have three independent variables (q, r, s) of the general form: A 1 G q q q + A 2 G q q r + A 3 G q r r + A 4 G q r r + A 4 G r r r + A 5 G q q s + A 6 G q s s + A 7 G q r s + A 8 G r r s + A 9 G r s s + A 10 G s s s + A 11 G q q + A 12 G q r + A 13 G q s + A 14 G r r + A 15 G r s + A 16 G s s + A 17 G q + A 18 G r + A 19 G s + A 20 G = 0 By using the assumption: G(q,r,s) = e ∫y(q)dq + ∫x(r)dr + z(s)ds , the above equation, according to this assumption, will be transformed to the non-linear second order ordinary differential equation with three independent functions. Thus, they have the general form: A 1 ( y ″ + 3 y y ′ + y 3 ) + A 2 ( x y ′ + x y 2 ) + A 3 ( y x ′ + y x 2 ) + A 4 ( x ″ + 3 x x ′ + x 3 ) + A 5 ( z y ′ + z y 2 ) + A 6 ( y z ′ + y z 2 ) + A 7 y x z + A 8 ( z x ′ + z x 2 ) + A 9 ( x z ′ + x z 2 ) + A 10 ( z ″ + 3 z z ′ + z 3 ) + A 11 ( y ′ + y 2 ) + A 12