А.Д. Алексеев, С.Н. Кудряшов - Уравнения с частными производными в примерах и задачах (1120422), страница 4
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cth2 x ∂∂xu2 − 2ycthx ∂x∂y+ y 2 ∂∂yu2 + 2y ∂u∂y = 0.222∂ u+ y 2 ∂∂yu2 + tg 3 x ∂u1.86. tg 2 x ∂∂xu2 − 2ytgx ∂x∂y∂x = 0.Ïðèâåñòè óðàâíåíèÿ ê êàíîíè÷åñêîìó âèäó â êàæäîé èç îáëàñòåé, ãäå òèïóðàâíåíèÿ ñîõðàíÿåòñÿ:21.87. y ∂∂xu2 +1.88.∂2u∂x2∂2u∂y 2= 0.2+ y ∂∂yu2 + α ∂u∂y = 0. α = const22221.89. y ∂∂xu2 + x ∂∂yu2 = 0.1.90. x ∂∂xu2 + y ∂∂yu2 = 0.1.91.
Íàéòè îáùåå ðåøåíèå óðàâíåíèé â çàäà÷àõ :à) 1.41, á) 1.42, â) 1.43, ã) 1.44, ä) 1.46, å) 1.48, æ) 1.49,ç) 1.50, è) 1.52, ê) 1.53, ë) 1.55, ì) 1.57, í) 1.59, î) 1.60,ï) 1.66, ð) 1.71.Íàéòè îáùåå ðåøåíèå ñëåäóþùèõ óðàâíåíèé:1.92.∂2u∂x222∂ u+ 4 ∂x∂y− 5 ∂∂yu2 +∂u∂x−∂u∂y= 0.321.93.2∂2u∂x2∂ u− sinx ∂x∂y+ (sinx − ctgx) ∂u∂x = 0.22∂ u1.94. 4x ∂∂xu2 − y ∂x∂y+ 7 ∂u∂x = 0.1.95.2∂2u∂x22∂ u− 4x ∂x∂y+ 4x2 ∂∂yu2 −2∂u∂x+ 2(x − 1) ∂u∂y = 0.2∂ u1.96. x ∂∂xu2 − y ∂x∂y+ 2 ∂u∂x = 0.22∂ u1.97.
2x ∂x∂y− y ∂∂yu2 − 3 ∂u∂y = 0.1.98.∂2u∂x222∂ u− 4x ∂x∂y+ 4x2 ∂∂yu2 −21 ∂ux ∂x= 0.2∂ u1.99. x ∂x∂y− 3y ∂∂yu2 − 5 ∂u∂y = 0.22∂ u1.100. x ∂∂xu2 − y ∂x∂y+ 4 ∂u∂x = 0.1.101.∂2u∂x222∂ u∂u− 2cosx ∂x∂y+ cos2 x ∂∂yu2 − 2 ∂u∂x + (2cosx + sinx) ∂y = 0.22∂ u1.102. x ∂∂xu2 + y ∂x∂y= 0.22∂ u1.103. 4x ∂x∂y− y ∂∂yu2 + 3 ∂u∂y = 0.222∂ u∂u1.104. x2 ∂∂xu2 − 2xy ∂x∂y+ y 2 ∂∂yu2 + x ∂u∂x + y ∂y = 0.221.105.
t2 ∂∂tu2 − x2 ∂∂xu2 = 0.3322∂ u1.106. 3x ∂∂xu2 − y ∂x∂y+ 4 ∂u∂x = 0.22∂ u1.107. 3x ∂x∂y− y ∂∂yu2 − 2 ∂u∂y = 0.22∂ u+ 5 ∂u1.108. 2x ∂∂xu2 − y ∂x∂y∂x = 0.1.109.∂2u∂x2∂ u+ 2sinx ∂x∂y− cos2 x ∂∂yu2 +1.110.∂2u∂x2∂ u+ 2sinx ∂x∂y− cos2 x ∂∂yu2 + cosx ∂u∂y = 0.1.111.∂2u∂x2∂ u− 2sinx ∂x∂y− (3 + cos2 x) ∂∂yu2 − cosx ∂u∂y = 0.1.112.∂2u∂x2∂ u∂u− 2sinx ∂x∂y− (3 + cos2 x) ∂∂yu2 + x ∂u∂x + (2 − sinx − cosx) ∂y = 0.22222∂u∂x22222+ (sinx + cosx + 1) ∂u∂y = 0.2∂ u1.113. x2 ∂∂xu2 − 3xy ∂x∂y+ 2y 2 ∂∂yu2 + 3y ∂u∂y = 0.222∂ u1.114.
3x2 ∂∂xu2 − 16xy ∂x∂y+ 16y 2 ∂∂yu2 + 15x ∂u∂x = 0.1.115.∂2u∂x21.116.∂2u∂x∂y2∂ u− 2 xy ∂x∂y+2− 2x ∂∂yu2 +x2 ∂ 2 uy 2 ∂y 2−∂u1x2 +y ( ∂x2 ∂ux ∂x+y 2 −x2 ∂uy 3 ∂y− x3 = 0.− 2x ∂u∂y ) + 1 = 0. ñëåäóþùèõ çàäà÷àõ òðåáóåòñÿ íàéòè ðåøåíèÿ óêàçàííûõ óðàâíåíèé ïðèçàäàâàåìûõ íà÷àëüíûõ óñëîâèÿõ :1.117. Óðàâíåíèå çàäà÷è 1.41; u|t=0 = 0,34∂u∂t |t=0= −x − 1.1.118. Óðàâíåíèå çàäà÷è 1.57; u|x=0 = 1,∂u∂x |x=0= 3y .1.119. Óðàâíåíèå çàäà÷è 1.53; u|x=0 = 2y ,∂u∂x |x=0= 5y .1.120.
Óðàâíåíèå çàäà÷è 1.46; u|x=0 = 2y ,∂u∂x |x=0= 4y .1.121. Óðàâíåíèå çàäà÷è 1.44; u|y=0 = 2x,∂u∂y |y=0= 3x + 1.1.122. Óðàâíåíèå çàäà÷è 1.49; u|y=0 = 3x,∂u∂y |y=0= 2x + 6.22∂2u∂y 2222222222∂ u1.123. 3 ∂∂xu2 − 2 ∂x∂y−= 0;∂ u1.124. 3 ∂∂xu2 − 5 ∂x∂y+ 2 ∂∂yu2 = 0;∂ u1.125. 2 ∂∂xu2 + 3 ∂x∂y− 5 ∂∂yu2 = 0;∂ u1.126.
3 ∂∂xu2 − 4 ∂x∂y− 7 ∂∂yu2 = 0;1.127.∂2u∂x222∂ u− 5 ∂x∂y + 6 ∂∂yu2 = 0;u|y=0 = f (x),∂u∂y |y=0= F (x).u|y=0 = f (x),∂u∂y |y=0= F (x).u|y=0 = f (x),∂u∂y |y=0= F (x).u|y=0 = f (x),∂u∂y |y=0= F (x).u|y=0 = f (x),∂u∂y |y=0= F (x).1.128. Óðàâíåíèå çàäà÷è 1.60; u|y=0 = ϕ(x),∂u∂y |y=0= ψ(x).1.129. Óðàâíåíèå çàäà÷è 1.92; u|y=0 = f (x),∂u∂y |y=0= F (x).1.130.
Óðàâíåíèå çàäà÷è 1.96; u|x=1 = y ,35∂u∂x |x=1= 2y 3 .1.131. Óðàâíåíèå çàäà÷è 1.97; u|y=1 = x4 ,1.132. Óðàâíåíèå çàäà÷è 1.100; u|x=1 = 3y 4 ,∂u∂y |y=1∂u∂x |y=01.133. Óðàâíåíèå çàäà÷è 1.102; u|x=1 = 2y + 1,1.134. Óðàâíåíèå çàäà÷è 1.103; u|y=1 = 4x3 ,1.135. Óðàâíåíèå çàäà÷è 1.107; u|y=1 = x,= 2y 5 .= y.∂u∂x |x=1∂u∂y |y=1∂u∂y |y=11.136. Óðàâíåíèå çàäà÷è 1.103; u|y=1 = x2 + 1,1.137. Óðàâíåíèå çàäà÷è 1.107; u|y=1 = 0,= 3x3 .= 8x.= 15x2 .∂u∂y |y=1∂u∂y |y=1= 4.√= 6x2 3 x.1.138. Óðàâíåíèå çàäà÷è 1.94; u|x=1 = 3y 5 ,∂u∂x |x=1= 2y 11 .1.139. Óðàâíåíèå çàäà÷è 1.99; u|y=1 = 4x4 ,∂u∂y |y=1= 2x8 .1.140.
Óðàâíåíèå çàäà÷è 1.106; u|x=1 = 4y 3 ,1.141. Óðàâíåíèå çàäà÷è 1.108; u|x=1 = y 2 ,∂u∂x |x=1∂u∂x |x=11.142. Óðàâíåíèå çàäà÷è 1.106; u|x=1 = 1 + y 4 ,1.143. Óðàâíåíèå çàäà÷è 1.108; u|x=1 = 0,36= y7.= 2y 7 .∂u∂x |x=1∂u∂x |x=1= y4.= y5.22∂ u1.144. 2x ∂x∂y− 3y ∂∂yu2 +22∂ u1.145. 3x ∂∂xu2 − 2y ∂x∂y+∂u∂y= 0;u|y=1 = 2 + 3x2 ,∂u∂x= 0;u|x=1 = y 5 + 3,∂u∂y |y=1∂u∂x |x=122∂u∂y= 0,u|y=1 = 3x2 + 2x,22∂u∂y= 0,u|y=1 = 3x2 + 1,∂u∂x= 0,u|x=1 = 1,∂ u− 4y ∂∂yu2 +1.146. 3x ∂x∂y∂ u1.147.
2x ∂x∂y− 5y ∂∂yu2 +22∂ u1.148. 4x ∂∂xu2 − 3y ∂x∂y−22∂ u1.149. 3x ∂∂xu2 − y ∂x∂y+2∂u∂x= 0,2∂ u1.150. 3x ∂x∂y− 2y ∂∂yu2 + 2 ∂u∂y = 0,22∂ u1.151. 4x ∂∂xu2 − y ∂x∂y+ 7 ∂u∂x = 0,u|x=1 = 3y,1.152. Óðàâíåíèå çàäà÷è 1.105; u|t=1 = 2x2 ,∂u∂y |y=1∂u∂x |x=11.153. Óðàâíåíèå çàäà÷è 1.113; u|y=1 = 1 + 2x2 ,√22∂u1.156. 4x2 ∂∂xu2 − y 2 ∂∂yu2 + 8x ∂u∂x + y ∂y = 0,37= 4 − x2= 2y 2= x2 .∂u∂y |y=11.154. Óðàâíåíèå çàäà÷è 1.114; u|y=1 = 5x4 − 3x2 ,1.155. Óðàâíåíèå çàäà÷è 1.105; u|t=1 = 2 x,= 5x + 2=1−y∂u∂y |y=1δuδt |t=1= 1 − 2x= 3y 3∂u∂x |x=1u|y=1 = 3x2 ,= 2y 2 − y∂u∂y |y=1∂u∂x |x=1u|x=1 = 3y 2 ,= x4= 4x2 .∂u∂y |y=1∂u∂t |t=1u|x=1 = 2y,=√= 10x4 − 9x2 .x.∂u∂x |x=1=022∂u1.157.
x2 ∂∂xu2 − 9y 2 ∂∂yu2 + 6x ∂u∂x + 6y ∂y = 0,∂u∂y |y=1u|y=1 = 3x,1.158. Óðàâíåíèå çàäà÷è 1.113; u|y=1 = 2x,1.159. Óðàâíåíèå çàäà÷è 1.114; u|x=1 = 2y 2 ,∂u∂y |y=1= x.∂u∂x |x=1=1.160. Óðàâíåíèå çàäà÷è 1.109; u|y=−cosx = 1 + 2sinx,1.163. Óðàâíåíèå çàäà÷è 1.112; u|y=cosx = 0,20 23y .∂u∂y |y=−cosx1.161. Óðàâíåíèå çàäà÷è 1.110; u|y=−cosx = 1 + cosx,1.162.
Óðàâíåíèå çàäà÷è 1.111; u|y=cosx = sinx,= x2 .∂u∂y |y=−cosx∂u∂y |y=cosx∂u∂y |y=cosx= sinx.= 0.= 21 ex .x= e 2 cosx.Óðàâíåíèå (1.1) ïðèâåäåíî ê êàíîíè÷åñêîìó âèäó ïðè ïîìîùè óêàçàííîéïîäñòàíîâêè (ξ, η). Çàâåðøèòü ïðèìåð, îïðåäåëèâ ÷àñòíîå ðåøåíèå, óäîâëåòâîðÿþùåå çàäàííûì íà÷àëüíûì óñëîâèÿì:1.164.∂2u∂ξ∂η− 2 ∂u∂ξ = 0,1.165.∂2u∂ξ∂η+1.166.1.167.ξ = 2x + 3y,η = 4x − 5y,u|x=0 = 1,∂u∂x |x=0= 2.= 0,ξ = 3x + 8y,η = 4x − 5y,u|x=0 = 5,∂u∂x |x=0= 7.∂2u∂ξ∂η− 4 ∂u∂ξ = 0,ξ = 3x + 7y,η = 4x − 5y,u|x=0 = 1,∂u∂x |x=0= 2.∂2u∂ξ∂η+ 3 ∂u∂ξ = 0,ξ = 3x − 4y,η = 5x + 6y,u|x=0 = 2,∂u∂x |x=0= 3.1 ∂u2 ∂ξ381.168.∂2u∂ξ∂η− 3 ∂u∂ξ = 0,ξ = 2x + 3y,η = 5x − 4y,1.169.∂2u∂ξ∂η− 2 ∂u∂η = 0,ξ = 5x − 6y,η = x + 2y,1.170.∂2u∂ξ∂η+= 0,ξ = 2x − 3y,η = 3x + 4y,u|x=0 = 2,∂u∂x |x=0= 1.1.171.∂2u∂ξ∂η+ 3 ∂u∂η = 0,ξ = 4x − 3y,η = 5x + 2y,u|x=0 = 3,∂u∂x |x=0= 5.1.172.∂2u∂ξ∂η− 3 ∂u∂η = 0,ξ = 3x − 4y,η = 3x + 5y,u|x=0 = y,∂u∂x |x=0= 1.1.173.∂2u∂ξ∂η+ 2 ∂u∂η = 0,ξ = 2x + 3y,η = 3x + 5y,u|y=0 = 2x,1.174.∂2u∂ξ∂η− 4 ∂u∂ξ = 0,ξ = 3x + y,1.175.∂2u∂ξ∂η= 0,1.176.∂2u∂ξ∂η+1 ∂u3η ∂ξ1.177.∂2u∂ξ∂η+2 ∂uη ∂ξ= 0,ξ = y 2 x3 ,1.178.∂2u∂ξ∂η+4 ∂uη ∂ξ= 0,ξ = xy 3 ,1.179.∂2u∂ξ∂η+1 ∂uη ∂ξ= 0,ξ = x3 y 2 ,1.180.∂2u∂ξ∂η+4 ∂uη ∂ξ= 0,ξ = y 3 x,1 ∂u4 ∂ηξ = x2 y 3 ,= 0,η = 2y − 5x,η = y,ξ = x2 y 3 ,η = x,η = y,η = x,η = y,39∂u∂x |x=0u|x=0 = 4,∂u∂x |x=1= 1.= 1.∂u∂y |y=0u|y=0 = 3x + 5,u|x=1 = 3y 3 + 5,η = x,∂u∂x |x=0u|x=0 = 1,∂u∂y |y=0= 3.= 4.= 3y + 1.u|y=1 = 2x,∂u∂x |y=1= 3x2 + 1.u|y=1 = 2x2 ,∂u∂y |y=1= 3x + 1.u|x=1 = 3y,∂u∂x |x=1u|y=1 = 2x3 ,u|x=1 = 1 + 2y,= 2 + 3y.∂u∂y |y=1= 3x.∂u∂x |x=1= 3y 2 .1.181∂2u∂ξ∂η+5 1 ∂u3 η ∂ξ= 0,ξ = x3 y 4 ,η = y,u|x=1 = 3y 5= 0,ξ = x3 y 4 ,η = y,u|x=1 = y1.182.∂2u∂ξ∂η+4 ∂u3η ∂ξ1.183.∂2u∂ξ∂η+3 ∂uη ∂ξ1.184.∂2u∂ξ∂η+9 ∂u2η ∂ξ1.1851.186.∂2u∂ξ∂η−∂2u∂ξ∂η+2 ∂uη ∂ξξ = x3 y 2 ,= 0,ξ = y 2 x3 ,= 0,ξ = x2 y 3 ,= 0,4 ∂u3η ∂ξη = y,η = x,ξ = x3 y 4 ,= 0,η = x,η = x,∂u∂x |x=1∂u∂x |x=1= 3y 4= 3y + 2u|x=1 = 3y 2∂u∂x |x=1= 3y + 2u|y=1 = x3∂u∂x |y=1= x2 − 2∂u∂x |y=1u|y=1 = x2 + 1u|y=1 = 4x2∂u∂x |y=1=x= 6x ñëåäóþùèõ çàëäà÷àõ òðåáóåòñÿ íàéòè ðåøåíèÿ çàäàííûõ óðàâíåíèé ïîçàäàííûì çíà÷åíèÿì ϕ(x),ψ(x) èñêîìûõ ðåøåíèé íà êóñêàõ ïàðû íåçàâèñè-ìûõ õàðàêòåðèñòèê:1.187.∂2u∂x2=∂2u∂y 2 ,u(x, y) = ϕ(x) íà y + x = 0,u(x, y) = ψ(x) íà y − x = 0,ϕ(0) = ψ(0).1.188.∂2u∂x222∂ u+ 6 ∂x∂y+ 5 ∂∂yu2 = 0,u(x, y) = ϕ(x) íà y − x = 0,u(x, y) = ψ(x) íà 5x − y = 0,ϕ(0) = ψ(0).1.189.∂2u∂x222∂ u+ 6 ∂x∂y+ 5 ∂∂yu2 = 0,u(x, y) = ϕ(x) íà y = 5x + 3,u(x, y) = ψ(x) íà y = x − 1,40ϕ(−1) = ψ(−1).1.190.∂2u∂x222∂ u− 6 ∂x∂y+ 8 ∂∂yu2 = 0,u(x, y) = ϕ(x) íà y + 4x = 0,u(x, y) = ψ(x) íà y + 2x + 2 = 0,ϕ(1) = ψ(1).22∂ u1.191.
3 ∂∂xu2 + 2 ∂x∂y−∂2u∂y 2= 0,u(x, y) = ϕ(x) íà x − y − 1 = 0,u(x, y) = ψ(x) íà x + 3y + 1 = 0,ϕ( 12 ) = ψ( 12 ).222∂ u+ 3 ∂∂yu2 = 0,1.192. 4 ∂∂xu2 − 8 ∂x∂yu(x, y) = ϕ(x) íà x + 2y + 1 = 0,u(x, y) = ψ(x) íà 3x + 2y + 2 = 0,ϕ(− 12 ) = ψ(− 12 ).222∂ u− 2 ∂∂yu2 = 0,1.193. 3 ∂∂xu2 + 5 ∂x∂yu(x, y) = ϕ(x) íà x + 3y + 2 = 0,u(x, y) = ψ(x) íà 2x − y − 1 = 0,ϕ( 17 ) = ψ( 17 ).222∂ u1.194.
25 ∂∂xu2 + 5 ∂x∂y− 2 ∂∂yu2 = 0,u(x, y) = ϕ(x) íà 2x − 5y − 4 = 0,u(x, y) = ψ(x) íà x + 5y + 3 = 0,ϕ( 13 ) = ψ( 13 ).1.195.∂2u∂x222∂ u+ 2 ∂x∂y− 8 ∂∂yu2 = 0,u(x, y) = ϕ(x) íà 4x − y + 3 = 0,u(x, y) = ψ(x) íà 2x + y − 4 = 0,ϕ( 16 ) = ψ( 16 ).411.196.∂2u∂x2+∂2u∂x∂y2− 6 ∂∂yu2 = 0,u(x, y) = ϕ(x) íà 2x + y + 1 = 0,u(x, y) = ψ(x) íà 3x − y − 2 = 0,ϕ( 15 ) = ψ( 15 ).222∂ u1.197. 2 ∂∂xu2 − 7 ∂x∂y− 4 ∂∂yu2 = 0,u(x, y) = ϕ(x) íà 4x + y + 1 = 0,u(x, y) = ψ(x) íà x − 2y + 4 = 0,ϕ(− 23 ) = ψ(− 23 ).1.198.∂2u∂x22∂ u+ 2x ∂x∂y−1 ∂2ux ∂y 2= 0, (x > 0),u(x, y) = ϕ(x) íà y − 1 = 0,u(x, y) = ψ(x) íà x2 − y = 0,ϕ(1) = ψ(1).1.199.∂2u∂x22+ 2x ∂∂yu2 = 0, (y > 0),u(x, y) = ϕ(x) íà y − x2 = 0,u(x, y) = ψ(x) íà x − 2 = 0,ϕ(2) = ψ(4).21.200.
2y ∂∂xu2 +∂2u∂x∂y= 0, (x > 0),u(x, y) = ϕ(x) íà y −√x = 0,u(x, y) = ψ(x) íà y − 2 = 0,ϕ(4) = ψ(4).1.201.∂2u∂x22− 4x2 ∂∂yu2 −1 ∂ux ∂x= 0, (x > 0),u(x, y) = ϕ(x) íà y − x2 = 0,u(x, y) = ψ(x) íà y + x2 − 2 = 0,ϕ(1) = ψ(1).1.202.∂2u∂x222∂ u1 ∂u∂u+ 2shx ∂x∂y− ∂∂yu2 + chx∂y − thx ∂x = 0,u(x, y) = ϕ(x) íà y − ex = 0,u(x, y) = ψ(x) íà y − e−x = 0,42ϕ(0) = ψ(0).2 ÓÐÀÂÍÅÍÈß ÃÈÏÅÐÁÎËÈ×ÅÑÊÎÃÎ ÒÈÏÀ. ÌÅÒÎÄ ÔÓÐÜÅ. äàííîì ðàçäåëå ìû ïðèâåäåì çàäà÷è, ðåøàåìûå ìåòîäîì Ôóðüå äëÿ óðàâíåíèé ãèïåðáîëè÷åñêîãî òèïà , ê êîòîðûì ñâîäÿòñÿ ôèçè÷åñêèå ïðîöåññû êîëåáàòåëüíîãî õàðàêòåðà.
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