The general solutions of some nonlinear second order PDEs.I. Two independent variables, constant parameters

The general solutions of some nonlinear se cond order PDEs. I. Tw o indep enden t v ariables, constan t parameters Yu. N. Koso vtso v Lviv Radio Engineering Resear c h Institute, Ukraine email: kosovt sov@escort.lviv.net Abstract In the first part of planned series of pap ers the formal general solutions to selection of 80 examples of different types of second order nonlinear PDEs in tw o ind e p enden t v ariables with constan t parameters are giv en. The main goal here is to show on examples the typ e s of solv able PDEs and what t heir general solutions look like. The solving strategy , used here, as a rule is the order reduction. The order reduction metho d is implemented in Maple pro cedure, whic h applicable to PDEs of different order with different n umber of indep enden t v ariables. S o me of given P DEs are solv ed by order lifting to PDEs, which are solv able by the subseq u en t order redu ct i on. 1 In tro duction Nonlinear partial differential equations (PDEs) play v ery imp ortan t role in many fields o f ma t hematics, physics, chemistry , and biolog y , and numerous applica- tions. Despite the fact that v arious methods for solving no nlinea r PDEs hav e bee n developed in 19-2 0 centuries [1]- [8 ], there exists a very disadv an tageous opinion that only a very small minority of nonlinea r second- a nd higher- order PDEs admit gener al solutions in c lo sed form (see, e.g ., p ercen tage of PDEs with general solutions in fundamental ha n dbo ok [9]). Nevertheless there exist some extensive no n trivial families for differe nt type s of no nlin ear PDEs whic h gener al solutions can be expressed in clo sed form and which seemingly ar e not describ ed in literature. In the first par t of planned series of pap ers the formal general solutions to selection of 80 examples of different types of second or der nonlinear P D Es in t wo indepe ndent v aria bles with c o nstan t parameter s are given. The main go a l here is to show on examples the t ypes o f solv a ble PDEs and what their genera l solutions lo ok like. The so lving strategy , used here , a s a r ule is the order reduction. The o rder reduction metho d is implemented in Maple pro cedure (see App endix), which applicable to PDEs of different order with different num ber o f indep enden t 1 v a riables [10]. So me of given PDEs ar e solv ed by or der lifting to PDEs, whic h are solv a ble by the subsequen t order reduction. I try to to follow the tradition when solv able PDEs ar e file d up to p oin t transformatio ns, deciding b et w een eq uiv alent PDEs v arian ts the mo s t short one. I would like to thank P rof. A.D. Poly anin for v aluable advice s in res ults presentation. 2 Equations of the F orm ∂ 2 w ∂ t∂ x = F ( w , ∂ w ∂ t , ∂ w ∂ x ) 2.1 ∂ 2 w ∂ t∂ x = 2 w  ∂ w ∂ t + 1  ∂ w ∂ x . General solution w ( x, t ) = − F ( t ) F ′ ( t ) + ( [ F ′ ( t )] 2 " G ( x ) − Z F ′′ ( t ) F ( t ) [ F ′ ( t )] 2 dt #) − 1 , where F ( t ) and G ( x ) are arbitra ry functions. 2.2  ∂ 2 w ∂ t ∂ x  2 =  ∂ w ∂ x  2 ∂ w ∂ t . General solution w ( t, x ) = − 4 F ′ ( t ) F ( t ) + G ( x ) + Z  F ′′ ( t ) F ′ ( t )  2 dt , where F ( t ) and G ( x ) are arbitra ry functions. 2.3  ∂ 2 w ∂ t∂ x − w ∂ w ∂ x  2 +  2 ∂ w ∂ t − w 2   ∂ w ∂ x  2 = 0 . General solution w ( t, x ) =  − 1 2 Z ( F ′ ( t )) 2 e F ( t ) dt + G ( x )  e − F ( t ) , 2 where F ( t ) and G ( x ) are arbitra ry functions. 2.4  ∂ w ∂ t + w  2 +  ∂ 2 w ∂ t∂ x + ∂ w ∂ x + ∂ w ∂ t + w  × exp ∂ 2 w ∂ t ∂x + ∂ w ∂ x + ∂ w ∂ t + w ∂ w ∂ t + w ! = 0 . General solution w ( t, x ) =  e − x Z e t F ( t ) e x p[ F ( t ) e − x ] dt + G ( x )  e − t , where F ( t ) and G ( x ) are arbitra ry functions. 2.5 ∂ 2 w ∂ t∂ x = a  ∂ w ∂ t + w  n − ∂ w ∂ x , where a 6 = 0, a nd n 6 = 1 are co nstan ts. General solution w ( t, x ) =  a 1 1 − n Z e t [ F ( t ) + (1 − n ) x ] 1 1 − n dt + G ( x )  e − t , where F ( t ) and G ( x ) are arbitra ry functions. 2.6 ∂ 2 w ∂ t∂ x − 1 w ∂ w ∂ t ∂ w ∂ x − c w ∂ w ∂ t − k w = 0 , where c, k ar e consta n ts. General solution w ( t, x ) = {− c Z exp[ x (1 − k t )] G ( x ) dx + F ( t ) } exp[ − x (1 − k t )] G ( x ) , where F ( t ) and G ( x ) are arbitra ry functions. 3 2.7 ∂ 2 w ∂ t∂ x = 1 w  ∂ w ∂ x + a  ∂ w ∂ t + bw ∂ w ∂ x , where a and b 6 = 0 ar e co nstan ts. General solution w ( t, x ) = − 1 b Z ∞ −∞ 1 ω  F ( ω ) ab exp  abt + ω 2 x ω  + G ( ω ) ω 2 exp  abx + ω 2 t ω  dω Z ∞ −∞  F ( ω ) exp  abt + ω 2 x ω  + G ( ω ) exp  abx + ω 2 t ω  dω , where F ( ω ) and G ( ω ) are arbitrary functions. 2.8 ∂ 2 w ∂ t∂ x −  1 w ∂ w ∂ t + b  ∂ w ∂ x − c w ∂ w ∂ t − cb = 0 , where b, c are constants. General solution w ( t, x ) =  − c Z exp[ − e bt G ( x )] dx + F ( t )  exp[ e bt G ( x )] , where F ( t ) and G ( x ) are arbitra ry functions. 2.9 ∂ 2 w ∂ t∂ x =  (2 w − a − 1) ( w − 1)( w − a ) ∂ w ∂ x + b ( w − a ) w − 1  ∂ w ∂ t , where a 6 = 1, a nd b are co nstan ts. General solution w ( t, x ) = G ′ ( x ) + abG ( x ) + aF ( t ) G ′ ( x ) + bG ( x ) + F ( t ) , where F ( t ) and G ( x ) are arbitra ry functions. 2.10 ∂ 2 w ∂ t∂ x = (2 k w − ak − c ) ( w − a ) ( kw − c ) ∂ w ∂ t ∂ w ∂ x + 2 b ( k w − c ) 2 ak − c , where a, b, c , and k ar e constants. 4 General solution w ( t, x ) = aF ′ ( t ) G ′ ( x ) − c [ F ( t ) − bG ( x )] 2 F ′ ( t ) G ′ ( x ) − k [ F ( t ) − bG ( x )] 2 , where F ( t ) and G ( x ) are arbitra ry functions. 2.11  ∂ 2 w ∂ t∂ x + ∂ w ∂ x  2 + a 2 "  ∂ w ∂ t + w  2 − b 2 #  ∂ w ∂ t + w  2 = 0 , where a 6 = 0, a nd b 6 = 0 are constants. General solution w ( t, x ) =  2 b Z F ( t ) e x p( abx + t ) 1 + F 2 ( t ) e x p( 2 abx ) dt + G ( x )  e − t , where F ( t ) and G ( x ) are arbitra ry functions. 2.12 ∂ 2 w ∂ t∂ x + ∂ w ∂ x + a "  ∂ w ∂ t + w  2 + b 2 #  ∂ w ∂ t + w  = 0 , where a 6 = 0, a nd b 6 = 0 are constants. General solution w ( t, x ) = ± b Z e t dt p F ( t ) e x p[( 2 ab 2 x ) − 1 + G ( x ) ! e − t , where F ( t ) and G ( x ) are arbitra ry functions. 2.13  w ∂ 2 w ∂ t∂ x − ∂ w ∂ t ∂ w ∂ x  exp " a ∂ 2 w ∂ t ∂x ∂ w ∂ x # = b ∂ w ∂ x , where a , and b are constants. General solution 5 w ( t, x ) =  − b Z exp [ aF ′ ( t ) + F ( t )] dt + G ( x )  e − F ( t ) , where F ( t ) and G ( x ) are arbitra ry functions. 2.14 w ∂ 2 w ∂ t ∂x ∂ w ∂ x − ∂ w ∂ t ! 2 = w  a ∂ w ∂ x − bw  , where a 6 = 0, a nd b are co nstan ts. General solution w ( t, x ) = F ( t ) exp    b a Z cos " t √ b 2 + G ( x ) #! − 2 dx    , where F ( t ) and G ( x ) are arbitra ry functions. 2.15 ∂ 2 w ∂ t∂ x + b ∂ w ∂ x = a exp  − ∂ w ∂ t − bw  , where a and b are constants. General solution w ( t, x ) =  Z ln [ ax + F ( t )] e bt dt + G ( x )  e − bt , where F ( t ) and G ( x ) are arbitra ry functions. 2.16 ( aw + 1)  ∂ w ∂ t + b  ∂ 2 w ∂ t∂ x = a ∂ w ∂ x  ∂ w ∂ t + b  2 − ( aw + 1) 3 , where a , b are constants. General solution 6 w ( x, t ) = −  Z  b ± p F ( t ) − 2 x  exp  ± a Z p F ( t ) − 2 x dt  dt + G ( x )  × exp  ∓ a Z p F ( t ) − 2 x dt  , where F ( t ) and G ( x ) are arbitra ry functions. 2.17 "  ∂ w ∂ t  2 − 2 aw #  w ∂ 2 w ∂ t∂ x − ∂ w ∂ t ∂ w ∂ x  2 = " w ∂ 2 w ∂ t∂ x ∂ w ∂ t + aw ∂ w ∂ x −  ∂ w ∂ t  2 ∂ w ∂ x # 2 , where a 6 = 0 is a consta n t. General solution w ( t, x ) =  − a 2 Z e F ( t ) F ′ ( t ) dt + G ( x )  e − F ( t ) , where F ( t ) and G ( x ) are arbitra ry functions. 2.18 ∂ 2 w ∂ t∂ x = w + 1 w ∂ w ∂ t ∂ w ∂ x + aw (1 − m ) e w (1 − m ) , where a , and m 6 = 1 a re constants. General solution in implicit form Z ∞ 1 e − ξw ( t,x ) ξ dξ = G ( x ) − Z [ a (1 − m ) x + F ( t )] 1 1 − m dt , where F ( t ) and G ( x ) are arbitra ry functions. 2.19 w  ∂ 2 w ∂ t∂ x + a ∂ w ∂ t  2 − ∂ w ∂ t  ∂ 2 w ∂ t∂ x + a ∂ w ∂ t  ×  2 aw + ∂ w ∂ x  + b  2 aw + ∂ w ∂ x  2 = 0 , where a and b are constants. 7 General solution w ( t, x ) =  b Z exp [ − ax − e ax F ( t )] F ′ ( t ) dt + G ( x )  exp [ e ax F ( t )] , where F ( t ) and G ( x ) are arbitra ry functions. 2.20  b 2 − 4 aw ∂ w ∂ x   aw 2 ∂ 2 w ∂ t∂ x − aw ∂ w ∂ t ∂ w ∂ x + b 2 ∂ w ∂ x  2 = b 2  aw 2 ∂ 2 w ∂ t∂ x − 3 aw ∂ w ∂ t ∂ w ∂ x + b 2 ∂ w ∂ x  2 , where a , and b 6 = 0 ar e co nstan ts. General solution w ( t, x ) = ±  b Z p F ′ ( t ) e aF ( t ) dt + G ( x )  e − aF ( t ) , where F ( t ) and G ( x ) are arbitra ry functions. 2.21 w ∂ 2 w ∂ t∂ x =  ∂ w ∂ x − aw  ∂ w ∂ t − b ∂ w ∂ x − cw 2 + abw , where a 6 = 0, b 6 = 0, a nd c ar e constants. General solution w ( t, x ) =  b Z exp  ct a + e − ax F ( t )  dt + G ( x )  exp  − ct a − e − ax F ( t )  , where F ( t ) and G ( x ) are arbitra ry functions. 2.22 ∂ 2 w ∂ t∂ x =  1 w ∂ w ∂ t + b  ∂ w ∂ x + c w ∂ w ∂ t + k w + cb , where b 6 = 0, c , and k 6 = 0 a re co ns t ants. General solution w ( t, x ) =  − c Z exp  k b 2 [ e bt G ( x ) + bx ]  dx + F ( t )  exp  − k b 2 [ e bt G ( x ) + bx ]  , where F ( t ) and G ( x ) are arbitra ry functions. 8 2.23 ∂ 2 w ∂ t∂ x = a w  ∂ w ∂ x  2 + 1 w ∂ w ∂ t ∂ w ∂ x +  b + c w  ∂ w ∂ x + c 2 aw ∂ w ∂ t + ( bw + c ) 2 4 aw , where a 6 = 0 a nd b, c are constants. General solution w ( t, x ) =  − c 2 a Z exp  1 2 a Z 2 dx t + G ( x ) + bx  dx + F ( t )  exp  − 1 2 a Z 2 dx t + G ( x ) + bx  , where F ( t ) and G ( x ) are arbitra ry functions. 2.24 a 2 w 4  c ∂ w ∂ t + ∂ w ∂ t ∂ w ∂ x − w ∂ 2 w ∂ t∂ x  2 = ac w + b + a w ∂ w ∂ x , where a 6 = 0, b , and c a r e consta n ts. General solution w ( t, x ) =  − c Z exp  − 1 4 a  − 4 bx + Z ( t + G ( x )) 2 dx  dx + F ( t )  × exp  1 4 a  − 4 bx + Z ( t + G ( x )) 2 dx  , where F ( t ) and G ( x ) are arbitra ry functions. 2.25 w ( ∂ w ∂ x + aw + b ) ∂ 2 w ∂ t∂ x − ∂ w ∂ t ∂ w ∂ x ( ∂ w ∂ x + aw + 2 b ) + b ( aw + b ) ∂ w ∂ t + cw 3 , where a, b, c ar e constants. 9 General solution w ( t, x ) = −  b Z exp  − Z  ± p G ( x ) + 2 ct − a  dx  dx + F ( t )  × exp  Z  ± p G ( x ) + 2 ct − a  dx  , where F ( t ) and G ( x ) are arbitra ry functions. 2.26 ∂ 2 w ∂ t∂ x − a w  ∂ w ∂ x  2 −  1 w ∂ w ∂ t + b + c w  ∂ w ∂ x − c 2 aw ∂ w ∂ t − kw − bc 2 a − c 2 4 aw = 0 , where a 6 = 0, b , c , a nd k are constants. General solution for b 2 − 4 k a 6 = 0 w ( t, x ) = − c 2 a ( Z exp " 1 2 a Z exp( t √ b 2 − 4 ak ) G ( x )( b + √ b 2 − 4 ak ) − √ b 2 − 4 ak + b 1 + exp( t √ b 2 − 4 ak ) G ( x ) dx # dx + F ( t ) ) × × exp " − 1 2 a Z exp( t √ b 2 − 4 ak ) G ( x )( b + √ b 2 − 4 ak ) − √ b 2 − 4 ak + b 1 + exp( t √ b 2 − 4 ak ) G ( x ) dx # , where F ( t ) and G ( x ) are arbitra ry functions. 2.27 ∂ 2 w ∂ x∂ t −  1 w ∂ w ∂ t + b  ∂ w ∂ x − c w ∂ w ∂ t − aw 2  c + kw + ∂ w ∂ x  − 1 − sw − bc = 0 , where a, b, c, k , s are consta n ts. General solution 10 w ( x, t ) = e − kx exp  Z W ( x, t ) dx   − c Z e kx exp  − Z W ( x, t ) dx  dx + F ( t )  , where F ( t ) and G ( x ) are arbitrary functions, and the W = W ( x, t ) is any ro ot of the transcendental eq uation Z W 0 ξ dξ ( s − bk ) ξ + bξ 2 + a = t + G ( x ) . 2.28 ( aw n + b ) ∂ 2 w ∂ t∂ x − k ( aw n + b ) (2 − m )  − ∂ w ∂ t  m =  anw ( n − 1) ∂ w ∂ x − c ( aw n + b )  ∂ w ∂ t , where a , b , c , k , n , and m 6 = 1 are constants. General solution in implicit form ( w = w ( t, x )) Z w s dξ aξ n + b + Z  F ( t ) e c ( m − 1) x − k c  1 1 − m dt + G ( x ) = 0 , where F ( t ) and G ( x ) are arbitra ry functions, s is a cons t ant. 2.29 ∂ 2 w ∂ t∂ x = w − n w ∂ w ∂ t ∂ w ∂ x + aw n ( m − 1) e w (1 − m ) , where a 6 = 0, n 6 = − 1, and m 6 = 1 are constants. General solution in implicit form w ( t, x ) n 2 exp  − w ( t, x ) 2  M n 2 , n +1 2 ( w ( t, x )) = (1) ( n + 1)[ a (1 − m )] 1 1 − m Z [ x + F ( t )] 1 1 − m dt + G ( x ) , where M p,q ( z ) is the Whittaker M function, F ( t ) and G ( x ) are ar bit rary func- tions. 11 2.30 ( aw 2 + bw + cb − ak ) ∂ 2 w ∂ t∂ x =  (2 aw + b ) ∂ w ∂ t + w 2 + 2 cw + k  ∂ w ∂ x , where a, b, c , and k ar e constants. General solution w ( x, t ) = F ( t ) − W ( t )  G ( x ) + Z W ( t ) [ a F ′ ( t ) + F ( t ) + c ] aF 2 ( t ) + bF ( t ) − ak + cb dt  − 1 , where F ( t ) and G ( x ) are arbitra ry functions, and W ( t ) = exp  Z [2 aF ( t ) + b ] F ′ ( t ) + F 2 ( t ) + 2 cF ( t ) + k aF 2 ( t ) + bF ( t ) − ak + bc dt  . 2.31 ( aw + b ) ∂ 2 w ∂ t∂ x =  ∂ w ∂ t  2 +  a ∂ w ∂ x + 2 cw + 2 k  ∂ w ∂ t + ( ak − bc ) ∂ w ∂ x + ( cw + k ) 2 , where a, b, c , and k ar e constants. General solution w ( x, t ) =  − k 2 Z F ( t ) + x k [ F ( t ) + x ] − b exp  Z ck [ F ( t ) + x ] + ak − b c k [ F ( t ) + x ] − b dt  dt + G ( x )  × exp  − Z ck [ F ( t ) + x ] + ak − b c k [ F ( t ) + x ] − b dt  where F ( t ) and G ( x ) are arbitra ry functions. 12 2.32 w ( aw − b )  ∂ 2 w ∂ t∂ x  2 = (2 aw − b ) ∂ w ∂ t ∂ w ∂ x ∂ 2 w ∂ t∂ x + c (2 aw − b ) 2  ∂ w ∂ x  2 , where a, b , and c are constants. General solution w ( x, t ) = F ( t ) + W ( t ) ×    G ( x ) − a Z W ( t ) h ( F ′ ( t )) 2 + F ′ ( t ) H ( t ) − 2 bcF ( t ) + 2 acF 2 ( t ) i F ( t ) [ a F ( t ) − b ] [ F ′ ( t ) + H ( t )] dt    − 1 , where F ( t ) and G ( x ) are arbitra ry functions, and W ( t ) = exp    Z [2 aF ( t ) − b ] h ( F ′ ( t )) 2 + F ′ ( t ) H ( t ) − 2 bcF ( t ) + 2 acF 2 ( t ) i F ( t ) [ a F ( t ) − b ] [ F ′ ( t ) + H ( t )] dt    , H ( t ) = ± q ( F ′ ( t )) 2 + 4 acF 2 ( t ) − 4 bcF ( t ) . 2.33 ∂ 2 w ∂ t∂ x + ak w exp  1 aw ∂ w ∂ t + c aw + b  − 1 w  ∂ w ∂ t + c  ∂ w ∂ x + m ∂ w ∂ t + abmw + mc = 0 , where a 6 = 0, b , c , k and m ar e constants. General solution w ( t, x ) =  − c Z exp  abt − a Z W ( t, x ) dt  dt + G ( x )  × exp  − abt + a Z W ( t, x ) dt  , 13 here W ( t, x ) is any solution of the following transcendental equation Z W ( t,x ) s dξ k e ξ + mξ + x + F ( t ) = 0 , where F ( t ) and G ( x ) are arbitra ry functions, s is any co nstan t. 2.34  ∂ 2 w ∂ t∂ x  2 + 2 a ∂ 2 w ∂ t∂ x ∂ w ∂ x + b  ∂ w ∂ t  3 + (3 abw − c 2 )  ∂ w ∂ t  2 + aw (3 abw − 2 c 2 ) ∂ w ∂ t + a 2  ∂ w ∂ x  2 + a 2 w 2 ( abw − c 2 ) = 0 , where a , and b 6 = 0, c 6 = 0 are constants. General solution w ( t, x ) = 4 c 3 e cx b Z F ( t )[ F ( t ) − 1] e at [ F ( t )( c + e cx ) − c ] 2 dt + G ( x ) ! e − at , where F ( t ) and G ( x ) are arbitra ry functions. 2.35 w  ∂ 2 w ∂ t∂ x  2 − ∂ w ∂ x  ∂ w ∂ t − k w + b  ∂ 2 w ∂ t∂ x −  ∂ w ∂ x  2  k ∂ w ∂ t − a + bk  + c "  ∂ w ∂ t + k w  2 + 2 b ∂ w ∂ t − 2 w (2 a − bk ) + b 2   ∂ w ∂ x − cw  = 0 , where a , b , c , and k are constants. General solution w ( t, x ) =  − Z  ae cx F ′ ( t ) + b  exp [ k t + F ( t ) e − cx  dt + G ( x )  × exp  − k t − F ( t ) e − cx  , 14 where F ( t ) and G ( x ) are arbitra ry functions. 2.36  k ∂ 2 w ∂ t∂ x + a ∂ w ∂ x  2 × "  k ∂ w ∂ t + aw  2 − 2 bm ∂ w ∂ t − 2 bcw # =  ∂ 2 w ∂ t∂ x  k 2 ∂ w ∂ t + ak w − bm  + ak ∂ w ∂ t ∂ w ∂ x + ∂ w ∂ x ( a 2 w − cb )  2 , where a , b , c , k , and m are constants. General solution w ( t, x ) = − b 2( am − ck ) F ( t ) Z [ cF ( t ) − mF ′ ( t )] 2 aF ( t ) − k F ′ ( t ) dt + G ( x ) ! , where F ( t ) and G ( x ) are arbitra ry functions. 2.37 ∂ 2 w ∂ t∂ x = − a w 2  ∂ w ∂ t  3 −  b w + c w 2   ∂ w ∂ t  2 +  1 w ∂ w ∂ x + ( ak − b )(3 ak + b ) 4 a − 2 bc 3 aw − c 2 3 aw 2  ∂ w ∂ t + c 3 aw ∂ w ∂ x − k ( ak − b 2 ) w 4 a + c ( ak − b )( 3 ak + b ) 12 a 2 − bc 2 9 a 2 w − c 3 27 a 2 w 2 , where a 6 = 0, b , c , a nd k are constants. 15 General solution w ( t, x ) = 1 3 a  − c Z exp  − 1 2 a Z ( ak − b ) V ( t, x ) − 2 ak V ( t, x ) + 1 dt  dt + G ( x )  × exp  1 2 a Z ( ak − b ) V ( t, x ) − 2 ak V ( t, x ) + 1 dt  , here V ( t, x ) = W  F ( t ) e x p  − (3 ak − b ) 2 x + 4 a 4 a  , where W ( z ) is the Lambert W function, F ( t ) and G ( x ) are ar bit rary functions. 2.38 bw ∂ 2 w ∂ t∂ x = a  ∂ w ∂ t  2 +  b ∂ w ∂ x + cw + 2 af  ∂ w ∂ t + bf ∂ w ∂ x + c 2 g (1 − g ) a w 2 + cf w + af 2 , where a 6 = 0 , b 6 = 0 , c 6 = 0, and g , f are constants. General solution w ( x, t ) = − 1 c + W ( t, x ) a × ( G ( x ) + Z  a ( af + g − 1) exp  c b [ x + 2 g F ( t )]  − ( af − g ) exp  c b [2 g x + F ( t )]  W ( t, x )  − a exp  c b [ x + 2 g F ( t )]  + exp  c b [2 g x + F ( t )]  dt ) , where F ( t ) and G ( x ) are arbitra ry functions, and W ( t, x ) = exp ( c a Z  a ( g − 1) exp  c b [ x + 2 g F ( t )]  + g exp  c b [2 g x + F ( t )]   a exp  c b [ x + 2 g F ( t )]  − exp  c b [2 g x + F ( t )]  dt ) . 16 2.39 ( a 2 bw 4 + ck 2 )  ∂ 2 w ∂ t∂ x  2 − 2 a 2 w 3 ∂ w ∂ x  2 b ∂ w ∂ t + k  ∂ 2 w ∂ t∂ x + 4 a 2 kw 2  ∂ w ∂ x  2 ∂ w ∂ t + 4 a 2 bw 2  ∂ w ∂ t  2  ∂ w ∂ x  2 = 0 , where a, b, c , and k ar e constants. General solution w ( x, t ) = F ( t ) − W ( t ) ( G ( x ) + a Z W ( t ) F ′ ( t )  aF 2 ( t ) + H ( t )  a 2 F 4 ( t ) + aF 2 ( t ) H ( t ) − 2 ck F ′ ( t ) dt ) − 1 , where F ( t ) and G ( x ) are arbitra ry functions, and W ( t ) = exp ( 2 a Z F ( t ) F ′ ( t )  aF 2 ( t ) + H ( t )  a 2 F 4 ( t ) + aF 2 ( t ) H ( t ) − 2 ck F ′ ( t ) dt ) , H ( t ) = ± q a 2 F 4 ( t ) − 4 cb [ F ′ ( t )] 2 − 4 ck F ′ ( t ) . 2.40 ( aw 6 + b )  ∂ 2 w ∂ t∂ x  3 − 2 w 5 ∂ w ∂ x  3 a ∂ w ∂ t + c   ∂ 2 w ∂ t∂ x  2 + 4 w 4  ∂ w ∂ x  2 ∂ w ∂ t  3 a ∂ w ∂ t + 2 c  ∂ 2 w ∂ t∂ x − 8 w 3  ∂ w ∂ x  3  ∂ w ∂ t  2  a ∂ w ∂ t + c  = 0 , where a , and b 6 = 0 , c 6 = 0 ar e co nstan ts. 17 General solution w ( x, t ) = F ( t ) − W ( t ) × ( G ( x ) + Z W ( t ) F ′ ( t )  12 bF ′ ( t ) F 2 ( t ) + c 3 P 2 ( t )  12 bF ′ ( t ) F 4 ( t ) − 6 bcF ′ ( t ) P ( t ) + c 3 F 2 ( t ) P 2 ( t ) dt ) − 1 , where F ( t ) and G ( x ) are arbitra ry functions, and W ( t ) = exp ( 2 Z F ( t ) F ′ ( t )  12 bF ′ ( t ) F 2 ( t ) + c 3 P 2 ( t )  12 bF ′ ( t ) F 4 ( t ) − 6 bcF ′ ( t ) P ( t ) + c 3 F 2 ( t ) P 2 ( t ) dt ) , P 3 ( t ) = − 108 b 2 c 6 [ F ′ ( t )] 2 " aF ′ ( t ) − √ 3 9 H ( t ) + c # , H ( t ) = ± s 27 a 2 b [ F ′ ( t )] 3 + 54 abc [ F ′ ( t )] 2 + 27 bc 2 F ′ ( t ) − 4 c 3 F 6 ( t ) bF ′ ( t ) . 2.41 4 a 2 b 2 f g 2 ( bcg w + ah )  a ∂ w ∂ t + cw  ∂ 2 w ∂ t∂ x =  bg ∂ w ∂ t − h  " 4 a 2 b 2 g 2 k 2  ∂ w ∂ t  2 +  4 a 3 b 2 cf g 2 ∂ w ∂ x + 4 ab 2 cg 2 k ( g + 3 k ) w + 4 a 2 bg hk ( g + k )  ∂ w ∂ t + 4 a 2 b 2 c 2 g 2 f w ∂ w ∂ x + b 2 c 2 g 2 ( g + 3 k ) w 2 + 2 abcg h ( g + k )( g + 3 k ) w + a 2 h 2 ( g + k ) 2  , where a 6 = 0, b 6 = 0, c , f 6 = 0, g 6 = 0, h and k are constants. General solution w ( t, x ) = − 1 bg E ( t, x )  h ( g + k ) Z E ( t, x )[ V ( t, x ) + 1 ] 2 k V ( t, x ) − g − k dt + G ( x )  , 18 here E ( t, x ) = exp  c ( g + 3 k ) a Z V ( t, x ) 2 k V ( t, x ) − g − k dt  , V ( t, x ) = W  − g + k 2 bf exp  − ( g + 3 k ) 2 [ F ( t ) + x ] 4 a 2 f  , where W ( z ) is the Lambert W function, F ( t ) and G ( x ) are ar bitrary functions. 2.42 V  ∂ w ∂ t + bw   ∂ 2 w ∂ t∂ x + b ∂ w ∂ x  + a = 0 , where a, b are co nstants and V ( z ) is a ny function. General solution w ( x, t ) =  Z W ( t, x ) e bt dt + G ( x )  e − bt , where W ( t, x ) is any solution of the following transcendental equation Z W ( t,x ) s V ( ξ ) dξ + a x = F ( t ) , F ( t ) and G ( x ) are arbitra ry functions, and s is an arbitrar y c onstant. 2.43 V ∂ 2 w ∂ t∂x (2 aw + b ) ∂ w ∂ x ! + ∂ w ∂ t = w ( aw + b ) ∂ 2 w ∂ t∂x (2 aw + b ) ∂ w ∂ x , where a, b are co nstants, and V ( z ) is a ny function. General solution w ( x, t ) = F ( t ) − exp  Z W ( t ) [2 aF ( t ) + b ] dt  ×  a Z W ( t ) e x p  Z W ( t ) [ 2 aF ( t ) + b ] dt  dt + G ( x )  − 1 , where W ( t ) is any solutio n of the follo wing transcendental equatio n V [ W ( t )] + F ′ ( t ) = W ( t ) F ( t ) [ aF ( t ) + b ] , F ( t ) and G ( x ) are arbitra ry functions. 19 2.44 ∂ 2 w ∂ t∂ x + ak w V  1 aw ∂ w ∂ t + c aw + b  − 1 w  ∂ w ∂ t + c  ∂ w ∂ x = 0 , where a 6 = 0, b , c a re constants, and V ( z ) 6 = 0 is any function. General solution w ( t, x ) =  − c Z exp  abt − a Z W ( t, x ) dt  dt + G ( x )  × exp  − abt + a Z W ( t, x ) dt  , here W ( t, x ) is any solution of the following transcendental equation Z W ( t,x ) s dξ V ( ξ ) + x + F ( t ) = 0 , where F ( t ) and G ( x ) are arbitra ry functions, s is any constant. 2.45 V w ∂ w ∂ x ∂ 2 w ∂ t∂x + b ∂ w ∂ x ! + 2 aw ∂ w ∂ x  ∂ w ∂ t + bw  ∂ 2 w ∂ t∂x + b ∂ w ∂ x = aw 2 , where a 6 = 0, b ar e constants, and V ( z ) is any function. General solution w ( x, t ) = F ( t ) − exp  Z 2 aF ( t ) W ( t ) dt − bt  ×  a Z 1 W ( t ) exp  Z 2 aF ( t ) W ( t ) dt − bt  dt + G ( x )  − 1 , where W ( t ) is any solutio n of the follo wing transcendental equatio n V  W ( t ) 2 a  + W ( t )[ F ′ ( t ) + bF ( t )] = aF 2 ( t ) , F ( t ) and G ( x ) are arbitra ry functions. 20 2.46 [( a 2 b 1 − a 1 b 2 ) w − a 1 b 3 + b 1 a 3 ] ∂ 2 w ∂ t∂ x =  ( a 2 b 1 − a 1 b 2 ) ∂ w ∂ t + a 2 b 3 − a 3 b 2  ∂ w ∂ x − ( b 1 ∂ w ∂ t + b 2 w + b 3 ) 2 V a 1 ∂ w ∂ t + a 2 w + a 3 b 1 ∂ w ∂ t + b 2 w + b 3 ! , where V 6 = 0 is an arbitrar y function, a i and b i are constants. General solution w ( t, x ) = exp  Z − a 2 + b 2 Y ( t, x ) a 1 − b 1 Y ( t, x ) dt   Z − a 3 + b 3 Y ( t, x ) a 1 − b 1 Y ( t, x ) exp  − Z − a 2 + b 2 Y ( t, x ) a 1 − b 1 Y ( t, x ) dt  dt + G ( x )  , where the function Y ( t, x ) is determined by the transcendental equation Z Y s dz V ( z ) = x + F ( t ) and F ( t ) and G ( x ) are arbitrar y functions, s is any consta nt. 3 Equations of the F orm ∂ 2 w ∂ t∂ x = F ( w , ∂ w ∂ t , ∂ w ∂ x , ∂ 2 w ∂ x 2 ) 3.1 ∂ 2 w ∂ t ∂ x = w ∂ 2 w ∂ t 2 . General solution w ( t, x ) = Z W s F ( ξ ) e − x/ξ dξ + G ′ ( x ) , 21 where W = W ( t, x ) is a solution of the following transcendent al equation t − Z W v ξ F ( ξ ) e − x/ξ dξ + G ( x ) = 0 and F ( ξ ) and G ( x ) are arbitrary functions, s and v are ar bitrary constants. 3.2 ( ∂ w ∂ x + w 2 ) ∂ 2 w ∂ t∂ x =  ∂ 2 w ∂ x 2 + 3 w ∂ w ∂ x + w 3  ∂ w ∂ t General solution in implicit form ( w = w ( t, x )): Z 1 − xw w s G ′ ( ξ ) ξ dξ + xG  1 − xw w  = F ( t ) , where F ( t ) and G ( z ) a re arbitra r y functions, s is an arbitrar y consta nt. 3.3 ∂ 2 w ∂ t∂ x + 1 w ∂ w ∂ x = 1 ∂ w ∂ x  ∂ w ∂ t − 1  ∂ 2 w ∂ x 2 General solution w ( t, x ) = G ′ [ W ( t, x ) + F ′ ( t )] + t , where W ( t, x ) is any solution of the following transcendental equation G [ W ( t, x ) + F ′ ( t )] + tW ( t, x ) = F ( t ) − tF ′ ( t ) + x , and F ( t ) and G ( z ) are arbitrar y functions . 3.4 ∂ 2 w ∂ t∂ x = w ∂ 2 w ∂ x 2 + n  ∂ w ∂ x  2 where n 6 = 0 is a consta nt . General solution w ( t, x ) = − 1 n Z W ( t,x ) s [ G ( ξ ) + t ] 1 − n n dξ + F ′ ( t ) , where W ( t, x ) is any solution of the following transcendental equation Z W ( t,x ) s [ G ( ξ ) + t ] 1 n dξ = x + F ( t ) , and F ( t ) and G ( ξ ) a re arbitrar y functions, s is any constant. 22 3.5 ∂ w ∂ x ∂ 2 w ∂ t∂ x = " ∂ 2 w ∂ x 2 + a  ∂ w ∂ x  2 # ∂ w ∂ t , where a is a constant. General solution in implicit form ( w = w ( t, x )): e aw F ( t ) = x + G ( w ) , where F ( t ) and G ( z ) a re arbitra r y functions. 3.6 w  ∂ w ∂ x + a  ∂ 2 w ∂ t∂ x =  w ∂ 2 w ∂ x 2 − a ∂ w ∂ x − a 2  ∂ w ∂ t , where a 6 = 0 is a consta nt . General solution w ( t, x ) = − ae aW ( t,x ) F ( t ) + G ′ [ W ( t, x )] , where W ( t, x ) is any solution of the following transcendental equation G [ W ( t, x )] = e aW ( t,x ) − x , and F ( t ) and G ( z ) are arbitrar y functions . 3.7 ∂ w ∂ x ∂ 2 w ∂ t 2 −  ∂ w ∂ t + a  ∂ 2 w ∂ t∂ x = ∂ w ∂ x " 2  ∂ w ∂ t  2 + 3 a ∂ w ∂ t + a 2 # , where a 6 = 0 is a consta nt . General solution in implicit form ( w = w ( t, x )) e at +2 w + F ( at + w ) e at + w + G ( x ) = 0 , where F ( z ) and G ( x ) are arbitrar y functions . 23 3.8 ∂ w ∂ x  ∂ 2 w ∂ t∂ x  2 + a  ∂ 2 w ∂ x 2  2 = ∂ w ∂ t ∂ 2 w ∂ x 2 ∂ 2 w ∂ t∂ x , where a is a constant. General solution w ( t, x ) = − a Z dt F ′ ( t ) + G [ x − F ( t )] , where F ( t ) and G ( z ) a re arbitra r y functions. 3.9 w  ∂ w ∂ x − aw + 1  ∂ 2 w ∂ t∂ x =  w ∂ 2 w ∂ x 2 − ∂ w ∂ x − ( aw − 1) 2  ∂ w ∂ t where a 6 = 0 is a consta nt . General solution w ( t, x ) = 1 a + e ax G ′ [ W ( t, x )] , where W ( t, x ) is an y solution of the following tra nscendental eq ua tion W ( t, x ) e − ax + aG [ W ( t, x )] = F ( t ) , and F ( t ) and G ( z ) are arbitr ary functions . 3.10 w  ∂ w ∂ x + w  ∂ 2 w ∂ t∂ x = " w ∂ 2 w ∂ x 2 +  ∂ w ∂ x  2 + w ( a + 2) ∂ w ∂ x + aw 2 # ∂ w ∂ t where a 6 = 1 is a consta nt . General solution in implicit form ( w = w ( t, x )): w a − 1 + ( a − 1) [ w a e ax F ( t ) + G ( x + ln w )] = 0 , where F ( t ) and G ( z ) a re arbitra r y functions. 24 3.11  ∂ w ∂ x + a  ∂ 2 w ∂ t∂ x = " ∂ 2 w ∂ x 2 − 2  ∂ w ∂ x  2 − 3 a ∂ w ∂ x − a 2 # ∂ w ∂ t where a 6 = 0 is a consta nt . General solution in implicit form ( w = w ( t, x )): e w + ae − ( w + ax ) F ( t ) + G  w a + x  = 0 , where F ( t ) and G ( z ) a re arbitra r y functions. 3.12  ∂ w ∂ x − ( w + a )(2 w + a )  ∂ 2 w ∂ t∂ x =  ∂ 2 w ∂ x 2 − (6 w + 4 a ) ∂ w ∂ x + ( w + a )(2 w + a ) 2  ∂ w ∂ t where a 6 = 0 is a consta nt . General solution w ( t, x ) = a a 2 e − [ W + ax ] F ( t ) − G ′  W a + x  e W − a 2 e − [ W + ax ] F ( t ) + G ′  W a + x  , where W = W ( t, x ) is an y solution of the following tra nscendental equa tion e W + ae − [ W + ax ] F ( t ) + G  W a + x  = 0 , and F ( t ) and G ( z ) are arbitrar y functions . 3.13 w  ∂ w ∂ x ∂ 2 w ∂ t∂ x + ∂ w ∂ t ∂ 2 w ∂ x 2  + ∂ w ∂ t ∂ w ∂ x  ∂ w ∂ x + aw  = 0 , where a is a constant. General solution in implicit form w ( t, x ) 3 = e − ax aW [ ae ax G ( W ) − 3 W ] [ W + F ( t )] , 25 where W = W ( t, x ) is an y solution of the following tra nscendental equa tion ae ax W [ W + F ( t )] G ′ ( W ) − 3 W 2 − ae ax F ( t ) G ( W ) = 0 and F ( t ) and G ( z ) are arbitrar y functions . 3.14  ∂ 2 w ∂ x 2 + a ∂ w ∂ x   ∂ 2 w ∂ t∂ x + a ∂ w ∂ t  = b where a, b are co nstants. General solution w ( t, x ) = ( ± Z e ax r [2 tW + G ( W )][2 bx + W ] W dx + F ( t ) ) e − ax , where W = W ( t, x ) is an y solution of the following tra nscendental equa tion W (2 bx + W ) G ′ ( W ) + 2 tW 2 − 2 bxG ( W ) = 0 and F ( t ) and G ( z ) are arbitrar y functions . 3.15  ∂ w ∂ x + aw   ∂ 2 w ∂ x 2 + a ∂ w ∂ x   ∂ 2 w ∂ t∂ x + a ∂ w ∂ t  = b where a, b are co nstants. General solution w ( t, x ) = ( (144 b ) 1 3 4 Z [ G ( W ) − xW − t ] 2 3 W 1 3 e ax dx + F ( t ) ) e − ax , where W = W ( t, x ) is an y solution of the following tra nscendental equa tion 2 W G ′ ( W ) − xW + t − G ( W ) = 0 and F ( t ) and G ( z ) are arbitrar y functions . 3.16 ∂ w ∂ x ∂ 2 w ∂ t∂ x = " ∂ w ∂ t + bw  ∂ w ∂ x  2 # ∂ 2 w ∂ x 2 + aw  ∂ w ∂ x  2 , where a , and b 6 = 0 ar e co nstants. General solution in implicit form 26 Z w ( t, x ) s dξ W ( t, ξ ) = x + F ( t ) , where W ( t, ξ ) = W is any solution of the following transc e ndent al equation (2 aξ + bW 2 )  G (2 aξ + b W 2 ) + t  − ln( − bW 2 ) + ln( ξ ) = 0 and F ( t ) and G ( z ) are arbitrar y functions , s is any constant. 3.17 ∂ 2 w ∂ t∂ x =  ∂ w ∂ t + aw  ∂ 2 w ∂ x 2  ∂ w ∂ x  − 1 − a ∂ w ∂ x + b  ∂ w ∂ t + aw  , where a, b are co nstants. General solution w ( t, x ) = e − at G  F ( t ) + e − bx  , where F ( t ) and G ( z ) a re arbitra r y functions. 3.18 ∂ 2 w ∂ t∂ x =  ∂ w ∂ t + aw  ∂ 2 w ∂ x 2  ∂ w ∂ x  − 1 + c ∂ w ∂ x + b  ∂ w ∂ x + k w  , where a , b , c 6 = − ( a + b ), and k are constants. General solution in implicit form Z w ( t, x ) s ( a + b + c ) dξ G [ ξ e at ] e ( b + c ) t + bk ξ + F ( t ) + x = 0 , where F ( t ) and G ( z ) a re arbitra r y functions, a nd s is an arbitra ry constant. 3.19 ∂ w ∂ x ∂ 2 w ∂ t∂ x = ∂ w ∂ t ∂ 2 w ∂ x 2 + aw m  ∂ w ∂ x  n , where a , m , and n 6 = 2 ar e constants. General solution in implicit form Z w ( t, x ) s dξ W ( t, ξ ) = x + F ( t ) , where W ( t, ξ ) = W is any solution of the following transc e ndent al equation W (2 − n ) + atξ m ( n − 2) = G ( ξ ) 27 and F ( t ) and G ( z ) are arbitrar y functions , s is any constant. 3.20 ∂ w ∂ x ∂ 2 w ∂ t∂ x =  ∂ w ∂ t + bw n  ∂ 2 w ∂ x 2 + aw m  ∂ w ∂ x  2 , where a , b 6 = 0, n 6 = 1 and m 6 = n − 1 are constants. General solution in implicit form Z w ( t, x ) s exp  aξ ( m − n +1) b ( m − n + 1)  G  ξ (1 − n ) − bt ( n − 1) b ( n − 1)  dξ = F ( t ) − x , where F ( t ) and G ( z ) a re arbitra r y functions, s is an y constant. 3.21 w  c ∂ w ∂ x + bw 3  ∂ 2 w ∂ t∂ x = " cw ∂ 2 w ∂ x 2 − c  ∂ w ∂ x  2 + w (2 bw 2 + ac ) ∂ w ∂ x + abw 4 # ∂ w ∂ t where a , b 6 = 0, and c a r e constants. General solution in implicit form ( w = w ( t, x )): √ 2 ex p h ac 2 bw 2 − ax i erf " √ 2 ac 2 w √ b # + G h c w 2 − 2 bx i = F ( t ) , where erf ( z ) is the er ror function, F ( t ) and G ( z ) ar e a rbitrar y functions. 3.22 ∂ w ∂ x ∂ 2 w ∂ t∂ x − ∂ w ∂ t ∂ 2 w ∂ x 2 + [ bV ( w ) + a ]  ∂ w ∂ x  2 + V ( w )[ bV ( w ) + a ] V ′ ( w ) ∂ 2 w ∂ x 2 = 0 , where a 6 = 0, a nd b are co nstants, V ( w ) 6 = cons t is any function. General solution in implicit form Z w ( t, x ) s V ( ξ ) G  e at [ bV ( ξ ) + a ] aV ( ξ )  dξ + x + F ( t ) = 0 , where F ( t ) and G ( z ) a re arbitra r y functions, s is an y constant. 28 4 Equations of the F orm ∂ 2 w ∂ t∂ x = F ( w , ∂ w ∂ t , ∂ w ∂ x , ∂ 2 w ∂ t 2 , ∂ 2 w ∂ x 2 ) 4.1  ∂ w ∂ x  2 ∂ 2 w ∂ t 2 −  ∂ w ∂ t  2 ∂ 2 w ∂ x 2 = 0 . General solution w ( t, x ) = F " ( xW − G ( W ) + t ) 2 W # , where W = W ( t, x ) is an y solution of the following tra nscendental equa tion G ( W ) − 2 W G ′ ( W ) + xW − t = 0 and F ( z ) a nd G ( z ) ar e arbitrary functions. 4.2 2 ∂ w ∂ t ∂ w ∂ x ∂ 2 w ∂ t∂ x −  ∂ w ∂ x  2 ∂ 2 w ∂ t 2 −  ∂ w ∂ t  2 ∂ 2 w ∂ x 2 = 0 . General solution in implicit form ( w = w ( t, x )): G ( w ) [ F ( w ) + x ] + t = 0 , where F ( z ) and G ( z ) ar e ar bitrary functions. 4.3  a ∂ w ∂ t + 2 ∂ w ∂ t ∂ w ∂ x + b ∂ w ∂ x  ∂ 2 w ∂ t∂ x = ∂ w ∂ x  a + ∂ w ∂ x  ∂ 2 w ∂ t 2 + ∂ w ∂ t  b + ∂ w ∂ t  ∂ 2 w ∂ x 2 , where a 6 = 0, a nd b are co nstants. General solution in implicit form ( w = w ( t, x )): a Z bt + ax a s dξ aG ( w + aξ ) − b + F ( w ) + t = 0 , where F ( z ) and G ( z ) ar e ar bitrary functions, s is any co ns tant. 29 4.4  ∂ 2 w ∂ t∂ x  2 − ∂ 2 w ∂ t 2 ∂ 2 w ∂ x 2 + a ∂ 2 w ∂ t 2 = 0 , where a is a constant. General solution w ( t, x ) = at 2 2 + tG ( W ) + F ( W ) + xW , where W = W ( t, x ) is an y solution of the following tra nscendental equa tion tG ′ ( W ) + F ′ ( W ) + x = 0 and F ( z ) a nd G ( z ) ar e arbitrary functions. 4.5  ∂ 2 w ∂ t∂ x  2 − ∂ 2 w ∂ t 2 ∂ 2 w ∂ x 2 + a ∂ 2 w ∂ t 2 ∂ w ∂ x = 0 , where a 6 = 0 is a consta nt . General solution w ( t, x ) = 1 a [ atW + aG ( W ) + F ( W ) e ax ] , where W = W ( t, x ) is an y solution of the following tra nscendental equa tion aG ′ ( W ) + e ax F ′ ( W ) + at = 0 and F ( z ) a nd G ( z ) ar e arbitrary functions. 4.6  ∂ 2 w ∂ t∂ x  2 − ∂ 2 w ∂ t 2 ∂ 2 w ∂ x 2 + a  ∂ w ∂ t  2 ∂ 2 w ∂ x 2 = 0 , where a 6 = 0 is a consta nt . General solution w ( t, x ) = 1 a { aF ( W ) + axW − ln [ G ( W ) + at ] } , where W = W ( t, x ) is an y solution of the following tra nscendental equa tion − G ′ ( W ) + axG ( W ) + aF ′ ( W )( G ( W ) + at ) + a 2 tx = 0 and F ( z ) a nd G ( z ) ar e arbitrary functions. 30 4.7  ∂ 2 w ∂ t∂ x  2 + a ∂ w ∂ x ∂ 2 w ∂ t∂ x +  b ∂ w ∂ x − ∂ 2 w ∂ x 2  ∂ 2 w ∂ t 2 = 0 , where a , and b 6 = 0 ar e co nstants. General solution w ( t, x ) = T e bx b + F ( T ) + Z t s W dξ , where W = W ( T , ξ ) is an y s o lution of the transcendental equation G ( W ) e − aξ + T = 0 , T = T ( t, x ) is any solution of the following transcendental equatio n b Z t s e aξ dξ G ′ [ W ( T , ξ )] = bF ′ ( T ) + e bx and F ( z ) a nd G ( z ) ar e arbitrary functions, s is any c onstant. 4.8 ∂ w ∂ x  ∂ 2 w ∂ t∂ x  2 + a ∂ 2 w ∂ t∂ x + ∂ w ∂ x  b − ∂ 2 w ∂ x 2  ∂ 2 w ∂ t 2 = 0 , where a , and b 6 = 0 ar e co nstants. General solution w ( t, x ) = ± ( T + 2 bx ) 3 2 3 b + F ( T ) + Z t s W dξ , where W = W ( T , ξ ) is an y s o lution of the transcendental equation G ( W ) + 2 aξ + T = 0 , T = T ( t, x ) is any solution of the following transcendental equatio n 2 b Z t s dξ G ′ [ W ( T , ξ )] = 2 bF ′ ( T ) ± √ 2 bx + T and F ( z ) a nd G ( z ) ar e arbitrary functions, s is any c onstant. 4.9  ∂ 2 w ∂ t∂ x  2 + a  ∂ w ∂ x  2 ∂ 2 w ∂ t∂ x + " b  ∂ w ∂ x  2 − ∂ 2 w ∂ x 2 # ∂ 2 w ∂ t 2 = 0 , where a , and b 6 = 0 ar e co nstants. 31 General solution w ( t, x ) = − ln( bx − T ) b + F ( T ) + Z t s W dξ , where W = W ( T , ξ ) is an y s o lution of the transcendental equation G ( W ) − aξ + T = 0 , T = T ( t, x ) is any solution of the following transcendental equatio n Z t s dξ G ′ [ W ( T , ξ )] − F ′ ( T ) = 1 b ( bx − T ) and F ( z ) a nd G ( z ) ar e arbitrary functions, s is any c onstant. 4.10  ∂ 2 w ∂ t∂ x  2 + a ∂ 2 w ∂ t∂ x − ∂ 2 w ∂ t 2 ∂ 2 w ∂ x 2 + b ∂ 2 w ∂ t 2 + c ∂ 2 w ∂ x 2 = bc , where a , b , and c are constants. General solution w ( t, x ) = ct 2 + bx 2 2 + xT + F ( T ) + Z t s W dτ , where W = W ( T , τ ) is any solutio n of the follo wing transcendental equatio n G ( W ) + T + aτ = 0 and T = T ( t, x ) is any solution of the following transcendental equation Z t s dτ G ′ [ W ( T , τ )] = F ′ ( T ) + x where F ( z ) and G ( z ) ar e ar bitrary functions, s is any co ns tant. 4.11 " ∂ 2 w ∂ t 2 ∂ 2 w ∂ x 2 −  ∂ 2 w ∂ t∂ x  2 # V ′  ∂ w ∂ x  = a ∂ 2 w ∂ t∂ x − b ∂ 2 w ∂ t 2 , where a , b are constants a nd V ( z ) 6 = const is an arbitrary function. General solution 32 w ( t, x ) = Z x s W dξ + Z t v H dτ + F ( T ) , where W = W ( T , ξ ) and H = H ( T , τ ) a re an y so lutions of the following tran- scendental equa tions V ( W ) − T + bξ = 0 , G ( H ) + T + aτ = 0 and T = T ( t, x ) is any solution of the following transcendental equation Z x s dξ V ′ [ W ( T , ξ )] − Z t v dτ G ′ [ H ( T , τ )] + F ′ ( T ) , here F ( z ) and G ( z ) are ar bitrary functions, s, v are any consta nt s. 4.12  ∂ 2 w ∂ t∂ x  2 − ∂ 2 w ∂ t 2 ∂ 2 w ∂ x 2 + b ∂ 2 w ∂ t 2 + a ∂ w ∂ x = ab , where a, b are co nstants. General solution w ( t, x ) = at 2 + bx 2 2 − tG ( W ) + xW + F ( W ) , where W = W ( t, x ) is an y solution of the following tra nscendental equa tion tG ′ ( W ) − F ′ ( W ) = x and F ( z ) a nd G ( z ) ar e arbitrary functions. References [1] G. Darb oux , Sur les equations aux der ivees partieles du second ordr e. Ann. Sci. Ecole Norm. Sup. 18 70, V.7,pp.163-17 3. [2] G. Darb oux, Leco ns sur la theor ie g enerale de s surfaces . V.I I. Paris: Her - mann, 1915 . [3] E.Go ursat, Lecons sur l’integration des equatio ns aux derivees partieles du second ordr e a deux v ariables indep endantes. V.I,I I. Paris: Her mann, 1896,1 898. [4] A.R. F orsyth, Theor y of differential equations. Part 4 . Partial differential equations, vol. 6 , Dov er P ress, New Y o rk, 1959 . [5] F. Calog ero, A so lv able nonlinear wav e equa tion, Studies Appl. Math. 7 0, pp. 189-1 99, 1 984. 33 [6] V.M. Boyko, W.I. F ushch ych , Low ering of o r der and general solutions of some class es of partial differential equations, Rep orts on Math. Phys., V. 41, No. 3, pp. 31 1 -318 , 1998. [7] S.P .Tsa rev, On Darboux in tegrable nonlinear partial differential equations, Pro c. Steklov Institute of Mathematics, v. 225 , p. 3 72-3 81, 199 9. [8] A. V. Zhiber , and V. V. Sokolov, Exact integrable Liouville type h yperb o lic equations [in Russian], Usp ekhi Mat. Nauk, V ol. 56, No. 1 , pp. 6 4-104 , 2001. [9] A. D. Poly anin and V. F. Zaitsev, Handb o ok of Nonlinear Partial Differ- ent ial Equa tions, Chapman & Hall/CRC Press , Bo c a Raton, 2004. [10] Y u.N. Kosovtsov, The decomp os ition metho d and Maple pro cedure for finding first integrals of nonlinear P DEs of a ny order with any num ber o f indep endent v ar iables. arXiv:0704.0 072 v1 [math-ph], ht tp://arxiv .org/ a bs/07 0 4.0072 , 20 07. 5 App endix. Maple pro cedure r e duc e PDE or der reduce PDE order:=pro c (pde,unk) lo cal B,W,N,NN,AR G,acargs,i,M,p de0 ,DN,IND,IND2,IND3,IND4,AR GS,SUB,SUB0, Z0,Barg s,EQS,XXX,WW,BB,PP ,p deI,IV,s,AN,NA; option ‘Copyright (c) 20 06-20 07 by Y uri N. Kos ovtso v. All rights reserved.‘; N:=PDET o ols[diffor der](op(1,[selectr emov e(has,indets(pde,function),unk)])); NN:=op(1,[selectremove(has,op(1,[selectremov e(has,indets(p de,function),unk)]),diff )]); AR G:=[op(unk)]; acarg s := {} ; for i from 1 to nops(ARG) do if PDET o ols[diffor der](NN,op(i,AR G))=0 then else acargs:= acarg s unio n { op(i,ARG) } fi; o d; acarg s :=conv ert(acarg s,list); M:=op(0,unk)(op(aca r gs)); if t yp e(p de,e quation)=true then pde0 :=lhs(subs(unk=M,p de))-rhs (s ubs (unk=M,p de)) else p de0:=subs(unk= M,pde) fi; DN:=[seq(seq(i,i=1..no ps(acarg s)),j=1..N)]; IND:=seq(op(combinat[c hoo se](DN,i)),i=1..N); IND2:=seq(op(combinat[c ho ose](DN,i)),i=1..N-2); IND3:=op(combinat[c hoo se](DN,N-1)); IND4:=op(combinat[c hoo se](DN,N)); AR GS:=op(unk),M,seq(conv ert(D[op(op(i,[IND2]))](op(0,unk)) (op(acarg s )),diff ),i=1..no ps([IND2])); SUB:= { M=W[0],seq (conv ert(D[op(op(i,[IND]))](op(0,unk)) (op(acarg s )),diff )=W[op(op(i,[IND]))],i=1..nops ([IND])) } ; 34 SUB0:= { W[0]=o p(0,unk)(op(ARG) ), seq(W[op(op(i,[IND]))]=subs(M=op(0,unk)(op(ARG)), conv ert(D[op(op(i,[IND]))](op(0,unk))(op(acargs)),diff )),i=1 ..nops([IND])) } ; Z0:=B(ARGS,seq(con vert(D[op(op(i,[IND3]))]( op(0,unk))(op(acar gs)),diff ), i=1..nops([IND3]))); Bargs:= op(indets(subs(SUB,Z0),name)); EQS:=convert(subs(SUB, { seq(diff(Z0,op(i,acargs ))=0,i=1..nops(acarg s)) } ),diff ); XXX:= { seq(W[op(op(i,[IND4]))],i=1..nops([IND4])) } ; WW:=select(type,indets(subs(SUB,p de0)), ’name’) in tersect { seq(W[op(op(i,[IND4]))],i=1..nops ([IND4])) } ; BB:=select(has,co mbinat[c hoose ](XXX, nops(aca rgs)),WW); PP:= {} ; NA:=0; pdeI:= { seq( { s ubs (subs(solve(EQS,op(i,BB)),subs(SUB,p de0))) } ,i=1..nops (BB )) } ; IV:= { seq(W[op(op(i,[IND4]))],i=1..no ps([IND4])) } ; for s from 1 to nops(p deI) do try AN:=[pds o lve(op(s,pdeI), { B } ,iv ars=IV)]; if AN=[NULL] then else NA:=1 fi; for i from 1 to nops(AN) do if op(0,lhs(op(i,AN)))=B then PP:=P P union { rhs(o p(i,AN)) } fi; o d; catch: end try; o d; PP:=subs(SUB0 ,P P); if NA=1 then if PP= then print(W ARNING(”SOLUTION EXISTS”)) fi;fi; RETURN(PP); end pro c: Calling Se quence : r e d uc e PDE or d er ( PDE , f ( ~ x )); PDE - partia l differe ntial equation; f ( ~ x ) - indeterminate function with its a rguments. Notice: The reduced PDE is B = 0. 35