А.Д. Алексеев, С.Н. Кудряшов - Уравнения с частными производными в примерах и задачах (1120422), страница 9
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ξ = xy ,η = 3yèëèξ = lny + 12 ln(x2 + 9), η = arctg x3 ,ýëëèïòè-÷åñêèé,∂2u∂ξ 21.39. ξ =+yx2∂2u∂η 2∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).− lnx, η = x − y ,1.40. ξ = xy + lnx,η = x + y,ãèïåðáîëè÷åñêèé,∂2u∂ξ∂η∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).ãèïåðáîëè÷åñêèé,∂2u∂ξ∂η∂u= F (ξ, η, u, ∂u∂ξ , ∂η ).Êàíîíè÷åñêèé âèä:1.41. ξ = x − t,η = x,∂2u∂ξ∂η1.42.
ξ = x + y ,η = y,∂2u∂η 2= 0.= 0.∂2u∂η 21.43. ξ = x + 2y ,η = x,1.44. ξ = 4x + y ,η = 2x + y ,1.45. ξ = x + y ,η = x,∂2u∂ξ 2−+= 0.1 ∂u2 ∂η∂2u∂ξ∂η∂2u∂η 2−+1 ∂u2 ∂ξ∂u∂η= 0.= 0.831.46. ξ = x − y ,∂2u∂ξ∂ηη = x + 3y ,+1.47. ξ = 2y − x,η = y,∂2u∂ξ 2+∂2u∂η 21.48. ξ = x + 3y ,η = x,∂2u∂η 2+1 ∂u3 ∂η1.49. ξ = x + 2y ,η = 3x + 2y ,1.50. ξ = y − 3x,η = x,1.51. ξ = x + 2y ,η = 3x,1.52. ξ = 2x − y ,η = x,1.53. ξ = x − 5y ,η = x − y,1.54. ξ = x + y ,∂2u∂η 2∂2u∂ξ 2∂2u∂η 2η = x,+∂2u∂ξ∂η∂2u∂η 2= 0.−1 ∂u4 ∂η= 0.= 0.+ 16 ( ∂u∂ξ +∂u∂η )= 0.= 0.+1 ∂u4 ∂η∂2u∂η 2++∂u∂η= 0.∂2u∂η 2∂u∂η+∂2u∂ξ 2η = x − y,1.55. ξ = 2x + 3y ,∂u∂η= 0.+∂2u∂ξ∂η+1 ∂u4 ∂η1 ∂u3 ∂η1.56. ξ = x + 2y ,η = 2x + y ,∂2u∂ξ 21.57. ξ = x + 3y ,η = 2x − y ,∂2u∂ξ∂η+= 0.+ 2 ∂u∂η = 0.= 0.∂2u∂η 2+∂u∂ξ84+ 2 ∂u∂η = 0.= 0.∂2u∂η 21.58. ξ = x + y ,η = y,1.59. ξ = x + y ,η = 3x − y ,∂2u∂ξ∂η+1 ∂u2 ∂ξ= 0.1.60.
ξ = x + 3y ,η = x + y,∂2u∂ξ∂η+1 ∂u2 ∂η= 0.1.61. ξ = xy ,η = xy ,1.62. ξ = ln(x +√1.63. ξ = y + x2 ,1.64. ξ = x3 y ,1.66. ξ = x,∂2u∂ξ∂ηη = y − x2 ,∂2u∂η 2∂2u∂ξ∂ηη = x,1.68. ξ = x2 − y 2 ,1.69. ξ = x + siny ,η = x2 ,η = x,∂2u∂η 2py 2 + 1),+∂2u∂η 2= 0.∂2u∂ξ∂η= 0.∂u∂η= 0.+∂2u∂ξ 2= 0.4 ∂u3η ∂η+η = x − y2,η = x + ey ,1.67. ξ = x + y 2 ,= 0.x2 + 1), η = ln(y +η = y,1.65. ξ = x + y 2 ,∂2u∂ξ∂η∂u+ (α + β) ∂u∂ξ + β ∂η + cu = 0.= 0.∂2u∂ξ 2∂2u∂η 2+∂2u∂η 2+1 ∂uξ−η ∂ξ+ η ∂u∂η = 0.85+1 ∂u2η ∂η= 0.= 0.1.70. ξ = x + y + cosx,1.71. ξ = x + cosy ,1.72. ξ = xy 3 ,1.74.
ξ = tgy ,∂2u∂η 2∂2u∂ξ 2η = lnx,∂2u∂η 2∂2u∂η 21.78. ξ = y 2 + 2ex ,∂2u∂ξ 2+η = y,η = x2 ,∂2u∂ξ∂η∂2u∂η 2∂2u∂η 2∂2u∂ξ 2+2ξ 2 ∂u1+ξ 2 ∂ξ= 0.= 0.η = ey − x,η = 4x,= 0.2ξ ∂uξ 2 +η 2 ∂ξ ;+η = y,1.76. ξ = ey − 2x,1.79. ξ = y + x2 ,=∂u+ 14 cos ξ+η2 ( ∂η −= 0.1 ∂uη ∂η+ξ ∂uη 2 ∂ξ−∂2u∂η 2η = x,1.75. ξ = x + cosy ,1.77. ξ = y 2 ,∂2u∂ξ∂ηη = x,η = x,1.73. ξ = xtg y2 ,∂2u∂ξ∂ηη = x − y − cosx,+−+= 0.1 ∂uξ ∂ξ= 0.1 ∂uη ∂η∂2u∂η 2= 0.+1 ∂u4η ∂ξ6η 2 ∂u4η 3 −ξ ∂η1.80. ξ = 4x3 − 3y 2 ,η = x,∂2u∂η 2+1.81. ξ = 2x + siny ,η = y,∂2u∂η 2= 0.86= 0.= 0.∂u∂ξ )= 0.1.82.
ξ = x + 2e−y ,∂2u∂ξ 2η = 2x,+∂2u∂η 2= 0.1.83. ξ = x + y + sinx,η = x − y − sinx,1.84. ξ = ytg x2 ,η = y,∂2u∂η 21.85. ξ = ychx,η = shx,1.86. ξ = ysinx,η = y,31.87. ξ = x,η = 32 y 2 ,−2ξ ∂uξ 2 +η 2 ∂ξ∂2u∂η 2∂2u∂η 2∂2u∂ξ 2++−∂2u∂η 23+∂2u∂ξ∂η3√η = 2 y,1.88. ξ = x,∂u∂ξ )+ η ∂u∂ξ ) = 0.= 0, y > 0,−1∂u(η−ξ) ( ∂ξ−∂u∂η )∂2u∂ξ 2= 0, y < 0.∂u2α−1 ∂u+ ∂η2 +η ∂η = 0, y > 0,√√α− 1∂u∂2u+ η−ξ2 ( ∂uξ = x − 2 −y , η = x + 2 −y , ∂ξ∂η∂ξ − ∂η ) = 0, y < 0.31.89. ξ = x 2 ,3η = y2(x > 0,y < 0),33ξ = (−x) 2 , η = (−y) 2 , (x < 0, y < 0),∂2u∂ξ 2∂u∂η 2++31 ∂u3ξ ∂ξ+1 ∂u3η ∂η= 0;323ξ = (−x) 2 − y 2 , η = (−x) 3 + y 2333(x < 0,3y > 0),ξ = x 2 − (−y) 2 , η = x 2 + (−y) 2 , (x > 0, y < 0),∂2u∂ξ∂η+1.90.
ξ =1 1∂u3 η 2 −ξ 2 (η ∂ξ√x, η =− ξ ∂u∂η ) = 0.√= 0.= 0.1 ∂u3η ∂ηξ = x − 32 (−y) 2 , η = x + 23 (−y) 2 ,∂u+ 41 sin ξ+η2 ( ∂η −= 0.1∂u1+η 2 (ξ ∂ξ2ξ ∂uη 2 ∂ξ∂2u∂ξ∂ηy (x > 0, y > 0),87ξ=√∂2u∂ξ 2−x, η =√−y , (x > 0, y > 0),21 ∂u+ ∂∂ηu2 − 1ξ ∂u∂ξ − η ∂η = 0;√√ξ = −x, η = y (x < 0, y > 0),√√ξ = x, η = −y , (x < 0, y > 0),∂2u∂ξ 2−∂2u∂η 2−1 ∂uξ ∂ξ−1 ∂uη ∂η= 0.1.91.à) u = ϕ(x − t) + ψ(x);á) u = ϕ(x + y) + ψ(2x + y);xâ) u = ϕ(x + 2y) + ψ(x + 2y)e 2 ;yã) u = ϕ(4x + y)ex+ 2 + ψ(2x + y);ä) u = ϕ(x − y) + ψ(x + 3y)ey−x4;xå) u = ϕ(x + 3y) + ψ(x + 3y)e− 3 ;æ) u = ϕ(x + 2y) + ψ(3x + 2y)ex+2y4;ç) u = ϕ(y − 3x) + ψ(y − 3x)e−x ;è) u = ϕ(2x − y) + ψ(2x − y)e−x ;ê) u = ϕ(x − 5y)e−x−y4+ ψ(x − y);xë) u = ϕ(2x + 3y) + ψ(2x + 3y)e− 3 ;ì) u = ϕ(2x − y) + ψ(x + 3y)ey−2x ;í) u = ϕ(3x − y) + ψ(x + y)e−3x−y2;î) u = ϕ(x + 3y) + ψ(x + y)e−x+3y2;ï) u = ϕ(x) + ψ(x − ey )e−x ;ð) u = ϕ(x + cosy) x1 + ψ(x).1.92.
ξ = x + y ,η = 5x − y ,∂2u∂ξ∂η−+ 16 ∂u∂η = 0, u = ϕ(x + y) + ψ(5x − y)e88x+y6.1.93. ξ = y ,∂2u∂ξ∂ηη = y − cosx,1.94. ξ = xy 4 ,∂2u∂ξ∂ηη = y,1.95. ξ = x2 + y ,η = x,1.96. ξ = xy ,η = y,1.97.ξ = xy 2 ,η = x,1.98.ξ = x2 + y ,−−1 ∂uη ∂ξ∂2u∂ξ∂ηη = x,∂2u∂η 2−= 0, u = ϕ(y) + ψ(y − cosx)ey .= 0, u = ϕ(x2 + y) + ψ(x2 + y)ex .∂u∂η= 0, u = yϕ(xy) + ψ(y).= 0, u = ϕ(xy 2 )x + ψ(x).1 ∂uη ∂ξ−∂u∂η= 0, u = ϕ(xy 4 )y 3 + ψ(y).3 ∂uη ∂ξ−∂2u∂η 2∂2u∂ξ∂η−1 ∂uη ∂η= 0, u = ϕ(x2 + y)x2 + ψ(x2 + y).1.99. ξ = x3 y ,η = x,∂2u∂ξ∂η−2 ∂uη ∂ξ= 0, u = ϕ(x3 y)x2 + ψ(x).1.100. ξ = xy ,η = y,∂2u∂ξ∂η−3 ∂uη ∂ξ= 0, u = ϕ(xy)y 3 + ψ(y).1.101.
ξ = y + sinx,1.102. ξ = xy ,1.103. ξ = xy 4 ,∂2u∂η 2η = x,η = y,η = x,∂2u∂ξ∂η−∂2u∂ξ∂η1.104. ξ = xy ,η = y,∂2u∂η 21.105. ξ = xt,η = xt ,∂2u∂ξ∂η+1 ∂uη ∂ξ+= 0, u = ϕ( xy )y + ψ(y).1 ∂uη ∂ξ1 ∂uη ∂η−2η− 2 ∂u∂η = 0, u = ϕ(ξ) + ψ(ξ)e .= 0, u = x1 ϕ(xy 4 ) + ψ(x).= 0, u = ϕ(xy)lny + ψ(xy).1 ∂u2ξ ∂η= 0, u = ϕ(xt) +89√xtψ( xt ).η = y,∂2u∂ξ∂η−1 ∂uη ∂ξη = xy 3 ,∂2u∂ξ∂η−1 ∂u3ξ ∂ηη = y,∂2u∂ξ∂η−3 ∂uη ∂ξ1.106. ξ = xy 3 ,1.107. ξ = x,1.108.
ξ = xy 2 ,1.109. ξ = x + y + cos,= 0, u = ϕ(xy 3 )y + ψ(y).1= 0, u = ϕ(x) + x 3 ψ(xy 3 ).= 0, u = ϕ(xy 2 )y 3 + ψ(y).η = x − y − cosx,1.110.ξ = x + y + cosx,∂2u∂ξ∂ηη = x − y − cosx,η−2+ ψ(η).+ 21 ∂u∂ξ = 0, u = ϕ(ξ)e∂2u∂ξ∂η= 0, u = ϕ(ξ) + ψ(η).1.111. ξ = 2x − y + cosx,η = 2x + y − cosx,∂2u∂ξ∂η= 0, u = ϕ(η) + ψ(ξ).1.112. ξ = 2x − y + cosx,η = 2x + y − cosx,∂2u∂ξ∂η+∂2u∂ξ∂η12xy ϕ(x y)1 ∂u4 ∂ξ= 0,ηu=ϕ(η) + ψ(ξ)e− 4 .1.113.
ξ = x2 y ,1η = xy ,1.114. ξ = xy 4 ,1.115. ξ = x2 + y 2 ,1.116. ξ = x,3η = xy 4 ,η = x,η = x2 + y ,+1 ∂uη ∂ξ= 0, u =+ ψ(xy).∂2u∂ξ∂η−2 ∂uη ∂ξ= 0, u = η 2 ϕ(ξ) + ψ(η).∂2u∂η 2−2 ∂uη ∂η− η 3 = 0, u = ϕ(ξ) + ψ(ξ)η 3 +∂2u∂ξ∂η+1 ∂uη ∂ξ+ 1 = 0, u = η1 ϕ(ξ) + ψ(η) −Çàäà÷à Êîøè.1.117. u = 21 t2 − xt − t.90η510 ;ξη2.71.118. u = − 73 e− 3 x (x + 3y + 3) + 71 (16 − 18x + 9y).11.119. u = e− 5 x (−25y + 5x − 110) + 27y − 27x + 110.x1.120. u = e− 3 (−12y − 4x − 54) + 14y − 14x + 54.y1.121. u = e 4 (12x + 3y + 12) − 10x − 5y − 12.y1.122. u = e 3 (6x + 4y + 24) − 3x − 6y − 24.1.123. u(x, y) =3f (x+y)+f (x−3y)4+14x+yRF (τ )dτ.x−3y1.124.
u(x, y) = 3f (x + y) − 2f (x + 23 y) + 21.125. u(x, y) =1.126. u(x, y) =2f (x+y)+5f (x− 52 y)73f (x−y)+7f (x+ 37 y)10++57x+yRx+ 23 yRF (τ )dτ.x− 52 y710x+ 37 yRF (τ )dτ.x−y1.127. u(x, y) = 3f (x + y3 ) − 2f (x + y2 ) + 6x+ y2R1x+yRF (τ )dτ.x+ y31.128. u = 23 e−y ϕ(x + y) − 21 ϕ(x + 3y) + 14 e−1.129. u = f (x + y) + 65 e− 6 (x+y)F (τ )dτ.x+yx+y2x+3yRx+y0z[F (z) − f (z)]e 6 dz .x− 51 y91z[3ϕ(z) + 2ψ(z)]e 2 dz .1.130. u = (x2 − 1)y 3 + y.1.131. u = x4 + 43 x3 (y 4 − 1).1.132.
u = 3y 4 + (x2 − 1)y 5 .1.133. u = 2y + 1 + ylnx.1.134. u = 4x3 + x(y 8 − 1).1.135. u = x + 3x2 (y 5 − 1).1.136. u = x2 + y 4 .√1.137. u = x2 3 x(y 6 − 1).1.138. u = 3y 5 + (x2 − 1)y 11 .1.139. u = 4x4 + x8 (y 2 − 1).1.140. u = 4y 3 + 21 y 7 (x2 − 1).1.141. u = y 2 + y 7 (x2 − 1).921.142. u = xy 4 + 1.1.143. u = (x − 1)y 5 .1.144. ξ = x3 y 2 ,η = x,∂2u∂ξ∂η+2 ∂uη ∂ξϕ(ξ)η2= 0,u=+ ψ(η),= 0,u = η1 ϕ(ξ) + ψ(η),u = x4 14 y 4 + 2 + 3x2 − 41 x4 .1.145. ξ = x2 y 3 ,η = y,∂2u∂ξ∂η+1 ∂uη ∂ξ4u = x2 y 2 − 43 yx 3 + y 5 + 3 − y 2 + 43 y.1.146. ξ = x4 y 3 ,η = x,∂2u∂ξ∂η+5 ∂u3η ∂ξ= 0,u=ϕ(ξ)5η3+ ψ(η),5u = 45 y 4 − xy 2 + 3x3 + 3x − 45 .1.147. ξ = x5 y 2 ,u=82558 xy1.148.
ξ = y 4 x3 ,η = x,∂2u∂ξ∂η+6+ 35 y 5 + 3x2 −η = y,∂2u∂ξ∂η3 ∂uη ∂ξ258x+= 0,u=ϕ(ξ)η3+ ψ(η),− 23 .5 ∂u3η ∂ξ= 0,u=ϕ(ξ)5η3+ ψ(η),7u = 76 y 3 x 2 + 1 − 76 y 3 .1.149. ξ = xy 3 ,η = y,∂2u∂ξ∂η+2 ∂uη ∂ξ= 0,u=ϕ(xi)η2+ ψ(η),2u = 32 x 3 − xy + 3y 2 − 23 + y.1.150. ξ = x2 y 3 ,η = x,∂2u∂ξ∂η+4 ∂u3η ∂ξ= 0,u = 2(y 2 − 1) + 51 x2 (1 − y 5 ) + 3x2 .93u=ϕ(ξ)4η3+ ψ(η),1.151. ξ = xy 4 ,η = y,∂2u∂ξ∂η−3 ∂uη ∂ξ= 0,u = η 3 ϕ(ξ) + ψ(η),1u = 8y 2 (1 − x− 4 ) + 3y.1.152. u =x2t+ x2 t2 .1.153. u = 1 + 2x2 y 2 .1.154. u = 5x4 y 2 − 3x2 y 3 .√1.155.
u = 2 xt.1.156. ξ = y 2 x,u=2y√xη=+1.157. ξ = yx3 ,y2x,∂2u∂ξ∂η−1 ∂u2η ∂ξ= 0,u = η 2 ϕ(xi) + ψ(η),∂2u∂ξ∂η+5 ∂u6η ∂ξ= 0,u=1√y lnx.xη=x3y ,7u = 13 y 3 x2 + 37 y 2 x +18 x7 y 13−15 ϕ(ξ)η6+ ψ(η),1 x23 y 23 .1.158. u = x(1 + y).81.159. u = (x4 + x 3 )y 2 .1.160. u = 1 + sin(x − y − cosx) + ey+cosx sin(x + y + cosx).1.161. u = 1 + cosx · cos(y + cosx).941.162. u = sinx · cos( y−cosx) + ex sh( y−cosx).22−1.163. u = 2e2x − y − cosxy − cosx· cosx · sin42.1.164.
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