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1、Please visit for more information_ 大学数学吧百度 6.2 1 1. . 21 2 tantan ttt x dxxttt x dxx ttt xxttt TTBudxCu dxdxu T uuBudxCu dxdxu uuTBC uuu dx a ubucuu 2. 2. 2 x dxxtt x dxx tt xxtt EuEuFdxdxu uuEF u dx a ufu 3. 3. t x dxxt t x dxx t t xxt kuAdxkuAdxHeAdxc Adx u uukH eu cdxc Duheu 6. 6. 21 2 22 2 2 tan
2、tan 1 2 1 2 tt x dxxtt l x xtt T xdxT xdxu T xdx uT x udxu T xx dxlx lxuu x 7. 7. 2 21 2 2 tantan tt x dxxtt l x xtt T xdxT xdxudx u T xdx uT x udxudxu T xgdxg lx g lx uuu x 9. 9. xt xt xxtxxttt xxtxxttt dI V xdxV xLdx VLI dt dVICV I xdxI xCdx dt VLILILCV ICVCVCLI 7.1 1. u(x,t) = 1 2 ( + ) + ( ) + 1
3、 2 () + (1) = 0, = 1, u(x,t) = 1 20 + 0 + 1 2 + = 1 2 ( + ) ( ) = (2) = sinx, = 2, u(x,t) = 1 2sin( + ) + sin ( ) + 1 2 2 + = + 1 2 3 3 | + (3) = 3, = , u(x,t) = 1 2 ( + )3+ ( )3 + 1 2 + = 1 2 ( + )3+ ( )3 + 1 2 2 2 | + (4) = , = 1, u(x,t) = 1 2 cos( + ) + (x at) + 1 2 1 + = cosxcosat + 1 2 ( + ) (
4、) 3. 3. - 11 ( , ) ()() 22 11 ( ( )() ()() 22 ) () x at x at u x txatxat a xatxatxa a txat xt d a 4. 4. 2 1 ( ,0) 1 ( ,0)sin 111 ( , )cos ()cos ()sin 22 11 cos ()cos ()cos ()cos () 22 cos () ttxx x t x at x at Va V V xAcoskxa CL I V xAkkx CCL V x tAk xatAk xatAkkx aCL Ak xatAk xatAk xatAk xat Ak xat
5、 2 1 ( ,0)cos 1 ( ,0)sin 111 ( , )cos ()cos ()sin 22 11 cos ()cos ()cos ()cos () 22 cos () ttxx x t x at x at Ia I C I xAkxa LCL V I xkAkx LL CC I x tAk xatAk xatAkkx LLaL CCCC Ak xatAk xatAk xatAk xat LLLL C Ak xat L 5. 5. 2 23 2 , 22 ( , ) x x xxx xx ttxx tt tt vvv uu xxx vvv u xxx va v v u x v x
6、t 6. 6. 22 2 22 2 22 22 2 22 2 1 11 let , 1 11 1 1 1 tt tt x x x tt x tt xx xuxu xhxaht v u hx v vhx u u vhxv hx u hx vhxvvxx xhahhx hx vhxvvx xhahhx vhx ha 22 2 tt ttxx hx v h va v 7. 7. 12 230 30 , 3,3 then 0, xxxyyy uuu u xyxy xy xy xyxy xy xy xyxy uuff 8. 8. 12 230 30 , 3,3 then 0, xxxyyy uuu u
7、xyxy xy xy xyxy xy xy xyxy uuff 10.10. 0 0 12 12 12 12 1 2 1 ( , )()() ( ,0)( )( )( ) ( ,0)( )( )( ) (0, )()()0.important 11 ( )( )( ) 2 11 ( )( )( ) 2 :0 1 ()( 2 t x x x x u x tf xatfxat u xf xfxx u xafxafxx utf atfat f xxd a fxxd a when xat f xatx 0 0 0 0 2 12 1 21 1 )( ) 11 ()()( ) 2 ( , )()() :0
8、 11 ()()( ) 2 11 ()().()( ) 2 ( x at x x at x x at x at x x atd a fxatxatd a u x tf xatfxat when xat f xatxatd a fxatf atxatxd a u 00 1111 , )()( )()( ) 22 x atat x xx x txatdatxd aa 13.13.See See 7.2 1. 1. 0 2 0 2 0 223 0 23 1 , 2 11 22 1 2 2 111 223 26 tx a t x a t x a t t x a t t t u x tad d a ad
9、 a xtxa tad xtxata xtat 0 0 0 2 0 2 1 ,8 2 1 8 2 1 16 2 1 8 2 4 yxy xy y xy xy y y u x yd d d yd y y 2. 2. 0 0 00 1 sinsin 2 1 2 1 2 2 111 I tx a t II x a t t tt tt t uxatxat ue d d a ax te d a xte dxe d xt exte xextx 8.1 1. 1. 2 * 2 (0, )( , )0 :. 0 (0)( )0.( .minus) ( )cossin (0)0 ( )sin0,. ( )sin
10、 0 ( ) txx nn n l nn uDu utu l t let uXT TX XTDX T DTX XX XX l X xAxBx XA n X lBlnN l n XxBx l n TDT l T tC e 2 2 2 1 11 0 0 11 ( , )sin ( , )( , ) ( ,0)( )( ,0)sin 2 ( )sin 2 ( , )( , )( , )( )sinsin Dt n Dt l nn n n nn nn l n n Dt l l n nn n ux tE ex l u x tux t n u xxuxEx l n Exxdx ll nn u x tux
11、tu x txxdx ex lll 2 2 (0, )( , )0 :. 0 (0)( )0.( .minus) ( )cossin (0)0 ( )sin0,. ( )cos 0 ( ) txx xx xx x x nn n uDu utu l t let uXT TX XTDX T DTX XX XXl X xAxBx XB n XlAlnN l n XxAx l n TDT l T t 2 2 0 00 0 00 00 0 ( , )cos ( , )( , ) ( ,0)( )( ,0)cos 12 ( ),( )cos 12 ( , )( , )( )( )cos n Dt l n
12、n Dt l nn n n nn nn ll n ll n n C e n ux tE ex l u x tux t n u xxuxEx l n Ex dx Exxdx lll n u x tux tx dxxxdx e lll 2 1 cos n Dt l n n x l 2 * (0, )( , )0 :. 0 (0)( )0.( .minus) ( )cossin (0)0 1 2 ( )cos0,. 1 2 ( )sin txx x x x nn uDu utu l t let uXT TX XTDX T DTX XX XXl X xAxBx XA n XlBlnN l n XxBx
13、 l 2 2 2 1 2 1 2 1 11 1 2 0 ( ) 1 2 ( , )sin ( , )( , ) 1 2 ( ,0)( )( ,0)sin 2 ( ) n Dt l nn n Dt l nn n n nn nn n n TDT l T tC e n ux tE ex l u x tux t n u xxuxEx l Ex l 2 0 1 2 0 11 1 2 sin 11 222 ( , )( , )( )sinsin l n Dt l l n nn n xdx l nn u x tux txxdx ex lll 2. 2. 2 2 2 2 * 2 (0, )(2, )0 ( ,
14、0)0 :. 0 (0)(2)0.( .minus) = cossin (0)0 ( )sin20,. 2 ( )sin 2 0 2 ttxx t nn ua u utut u x let uXT TX XTa X T a TX XX XX X AxBx XA n X lBnN n XxBx n a TT 1 1 212 001 ( )cossin 22 (0)0,0 sincos 22 ( , )( , ) ( ,0)( )sin 2 2 ( )sinsin(2)sin 2222 22 sincoscosco 222 nnn n nn n n n n n n an a T tCtDt TD
15、nn a uExt u x tux t n u xxEx nnn Exxdxhxxdxhxxdx nnn xxxdxxx nn 122 011 s 2 22 cossin 22 22222 cossin2coscossin 22222 22222 cossin2coscos2coss 222 n n xdx nn xxx nn nnnnn Ehxxxhxhxxx nnnnn nnn hhnhn nnnnn 22 22 1 2 incossin 22 8 sin 2 8 ( , )sinsincos 222 n nn n n hn n hnnn a u x txt n 3. 3. 2 2 2*
16、2 :. 0 (0)( )0.( .minus) cossin (0)0 ( )sin0,. ( )sin 0 (0)0 ( )cossin (0)0,0. ( nn nnn n let uXT TX XTa X T a TX XX XX XAxBx XA XBnnN XxBnx TnaT T T tCnatDnat TD T t 1 1 1 )cos sincos ( , ) ( ,0)3sinsin 3,0.(1) ( , )3sin cos n nn n n n n n Cnat uEnxnat u x tu u xxEnx EEn u x txat 2 2 2 2 :. 0 (0)(
17、)0.( .minus) cossin (0)0 1 ( )cos0,. 2 1 ( )sin 2 1 0 2 (0)0 ( x x nn n let uXT TX XTa X T a TX XX XX XAxBx XA XBnnN XxBnx TnaT T T 0 3 0 3 0 11 )cossin 22 (0)0,0. 1 ( )sin 2 11 sinsin 22 ( , ) 1 ( ,0)sin 2 2123*( 1) sin* 2 1 2 nn n nn nn n n n n n n tCnatDnat TC T tDnat uEnxnat u x tu u xxEnx Exnxd
18、x n 2 22 2 22 0 2 1 2 23*( 1)211 ( , )*sinsin 22 11 22 n n n u x tnxnat nn 0 0 2 * 2 : 4 ( ,0) (0, )(1, )0 :. 4 4 0 (0)(1)0.( .minus) cossin (0)0 (1)sin0,. ( )sin 40 ( ) txx nn n let uvN vv v xN vtvt let vXT TX XTX T TX XX XX XAxBx XA XBnnN XxBn x TnT T tC 2 2 2 4 4 1 0 1 1 0 0 0 4 0 1 0 sin ( , ) (
19、 ,0)sin 2 2sin(cos1) 2 ( , )(cos1)sin ( , )( , ) nt n nt nn n n n n n nt n e uEn xe v x tu v xNEn x N ENn xdxn n N v x tnn xe n u x tv x tN 2 * 2 :. 0 (0)(1)0.( .minus) cossin (0)0 (1)sin0,. ( )sin 0 (1)0,(0)0 ( )sh(1) sinsh(1) ( nn nn nn nn let uXY YX XYX Y YX XX XX XAxBx XA XBnnN XxBn x YnY YY YyD
20、ny uEn xny u 1 1 1 33 0 3 3 , )( , ) ( ,0)(1)sinsh() 2 1 ( 1)4 1 ( 1) ( 1)( 1) sh()2(1)sin2 4 1 ( 1) sh 4 1 ( 1) ( , ) s n n n n nn nn n n n n x tux t u xx xEn xn Enx xn xdx nn nn E nn u x t n 1 sinsh(1) hn n xny n Please visit for more information_ 大学数学吧百度 4. 4. 2 2 2 2 2 Assume , we obtain cossin
21、000 0, 0,. si . n . . nn tttxx nnnn uauc u uXT XTaXTc X T TaTX c TX XAxBx XA n X n XB lln l n c TaTTT x ee l l 5. 5. 2 2 2 2 (0, )( , )0 ( ,0)0 ( ,0) :. 0 (0)( )0.( .minus) cossin (0)0 1 ( )cos0,. 2 ttxx x t x ua u utu l t u x Q ux ES let uXT TX XTa X T a TX XX XX l XAxBx XA X lBlnnN l 2 1 ( )sin 2
22、1 0 2 11 cossin 22 (0)0 1 cos 2 11 sincos 22 11 ( , )sincos 22 nn nnn nn nn nn n XxBnx l a TnT l aa TCntDnt ll TD a TCnt l a uEnxnt ll u x tEnxn l 0 0 0 0 2 2 11 ( ,0)cos 22 1 sin 1212122 cos* 1122 22 12 * 1 2 2 ( , ) n xn n l n l n n n a t l Q uxnEnx ESll nx lQQQ nEnxdx llESllESlES nn ll Ql E ES n
23、uu x t 2 0 2 111 *sincos 22 1 2 n n Qla nxnt ESll n 6. 6. 2 2 2 2 (0, )( , )0 ( ,0)0 ( ,0) :. 0 (0)( )0.( .minus) cossin (0)0 1 ( )cos0,. 2 ( ttxx x t x n ua u utu l t u x ux let uXT TX XTa X T a TX XX XX l XAxBx XA X lBlnnN l X 1 )sin 2 n xBnx l 2 0 1 0 2 11 cossin 22 (0)0 1 cos 2 11 sincos 22 11 (
24、 , )sincos 22 1 ( ,0) nnn nn nn nn n n x a TnT l aa TCntDnt ll TD a TCnt l a uEnxnt ll a u x tEnxnt ll uxn 0 0 0 2 2 2 2 1 cos 22 1 sin 1212122 cos* 1122 22 1 2* 1 2 11 ( , )2*sin 2 1 2 n n l n l n n n n Enx ll nx l nEnxdx lllll nn ll El n uu x tln n 0 1 cos 2 n a xnt ll 7. 7. 2 2 2 2 2 22222 :. 0,(
25、0)(1)0,sin, 0, (0)(1)0,sin, 0,(0)(1)0,sin, ()0 nn mm pp let uXYZT XYZTaX YZTXY ZTXYZ T TXYZ a TXYZ XXXXXAn xn YYYYYBm ym ZZZZZCp zp TanmpT 222222 222 222 222 1 ( )cossin (0)0,0 ( )cos ( , , , )sinsinsincos ( , , , )sinsinsincos ( , , ,0)si nmpnmpnmp nmp nmpnmp nnmp nmp nmp TtDnmpatEnmpat TE TtDnmpat
26、 ux y z tFn xm yp znmpat u x y z tFn xm yp znmpat u x y z 1 111 nsinsinsinsinsin 1,0,(111) ( , , , )sinsinsincos 3 nmp n nmp xyzFn xm yp z FFnmp u x y z txyzat 8. 8. 0 0 0 (0, )(0, )0 ,: ( , )0( , )0 ( ,0)( ,0)0 let: =. 0 (0)( )0.( .minus) ( )cossin (0)0 ( )co txxtxx xx uDuvDv utuvt let uvu v l tu l
27、 t v xuu x v XT TX XTDX T DTX XX XX l X xAxBx XA X lB 2 2 2 2 1 2 1 2 1 2 1 s0,. 2 1 sin 2 1 0 2 ( ) 1 sin 2 1 ( , )sin 2 nn a nDt l nn a nDt l nn n n lnnN l XBnx l a TnDT l T tC e vEnxe l v x tEnxe l 2 2 0 0 0 0 0 0 1 2 0 0 0 1 ( ,0)sin 2 221 sin 12 2 21 ( , )sin 12 2 ( , )( , ) a Dt l n n n l n a
28、nDt l n v xuEnx l u Eunxdx ll nl u v x tnxe l nl u x tv x tu 9 9. . (4) 2 2 (4) 4444 4 * 4 2 2 :. 0 0 (0)(0)0 ( )( )0 cossinsinhcosh ( )sin0,. ( )sin 0 ( ) nn n let uXT TX XTa XT a TX XX XX X lXl XAxBxCxDx n X lBlnN l n XxBx l n TaT l T t 22 22 22 1 0 1 cossin cossinsin ( , )cossinsin 2 ( ,0)( )sin,
29、( )sin nn nnn nn n l nn n t nn EatFat ll nnn uEatFatx lll nnn u x tEatFatx lll nn u xxEx Exxdx lll u 22 0 1 22 0 2 0 1 2 ( ,0)( )sin,( )sin 2 ( )sin 2 ( , )( )sincossinsin l nn n l l n nnnn xxFax Faxxdx lllll n xxdx nnnn ll u x txxdxatatx lllll n a l 11.11. 22 2 2 2 22 222 22 22 2 2 2 22 cossin 0,0
30、2 0,2, 8 sin 2 1 ,0sinsin 2 1 ,0 1 , n n n E t i nn E t i nn n n i tx XT i XTX T TX iE TX EE XAxaBxa XaA En X aanE a TC e n D exa a n xDxaxa aaa DD a x te a 2 sin E t i xa a 8.2 3. 3. 1. 2 2 2 * 2 1 2 11 2 :. 0 (0)0( )sin. ( )0 ( , )sin sinsin si ttxx nn ntxx n nn nn nn ua u let uXT TX XTa X T a TX
31、XX n XXxBxnN l X l n u x tTxua uAx l nn an TxTxAx lll n a TT l 1 2 0 2 2 2 n 22 sin1 2 1 cossin 2 1 (0)0 (0)0 2 1 cos1 2 1 ( , ) n l n nn n nnn n nn nn n n n n xAx l n anAl TTAxxdx llln Al n an a n TCtDt ll n a l Al n TC n a l n a TD l Al n a n Tt l n a l Al n u x t 2 1 cos1sin n n n an tx ll n a l
32、2. 2 2 2 2 2 0 0 0 :. 0 sin (0)( )0 1 ( )0,. 2 1 sin 2 1 sinsin 2 1 sin 2 txx nn ntxx n n n ua u let uXT TX XTa X T a TX XX XBx XX l X lnnN l XBnx l uTnxua uAt l Tnx l 2 0 2 0 2 0 11 sinsin 22 11 sinsin 22 122 sin sin 12 2 Asume:cossin n n nn n l nn nnn a TnnxAt ll a TTnnxAt ll aAt TTnAtdx ll n TXtY
33、t 2 2 4 2 2 12 sin sincoscossin 12 2 2 1 2 1 1 0 2 2 12 2 12 1 2 nnnn n nn nn n aAt XtYtnXtYt l n A n X a a YnX n l l aA A nYX l n n Y 2 2 4 2 2 1 2 44 22 1 2 2 1 2 221 112 22 ( )cos 11 22 a nt l nn a n l a n l AAa n l nn T tD et aa nn ll 2 44 22 1 2 4 2 sin 22 11 22 (0)0, 11 22 221 112 22 ( )1*cos
34、1 2 nn a nt l n t AA nn TDD aa nn ll AA n nn T tet a n l 2 2 4 2 21 2 4 2 4 2 sin 1 2 211 *1cossin 12 1 2 2 21 ( , )* 1 1 2 2 a nt l a l t a n l Aa etnt l a n n l A u x t a n n l 2 21 2 0 11 1cossinsin 22 a nt l n a etntnx ll 8.5 1 1 2 22 2 2 2 2 2 0 0 :. 0 ( )cossin ( )1 0 ( , )cossin :, ( , )cossi
35、n ( , nnn nn nnn nn nnnn n n nn n let uR ddRR d dd d n AnBnd RCDddRn R d u RAnBnCD whena uAnBn u a 0 1 0 0 1 )coscossin ,0 ( , )cos :, ( , )cossin ( , )coscossin ,0 ( , )cos n nn n n n nn n n nn n n AAnBna AaA B A u a whena uAnBn u aAAnBna A A B a Aa u 2 2. . 2 0 0 0 ( )( )0 ( )cos ( )cos0, ( )cos (
36、 ) ( , )cos ( , )cos( ) 2 ( )cos 2 ( , ) nn n nn n n n n n n n n B n B n B RC n uC n u aC af n C afd u 0 ( )coscos n n n ann fd 3. 3. 2 2 0 2 0 2 0 2 22 : 0 ( , ) ( ) ( , )cossin ( , )cossin ,0 ( , ) ( , ) n n n nn n n nn n n let uv v v aa RC vAnBn v aaAnBna AaB va ua 4. 4. 0 0 0 10 1 01 12 011 21 0
37、 ( , )cossin lim ( , )coscossin ,0 ( , )cos ( , )cos0, ( , )cos nn nnnn n n nn r n nn u rAnBnC rD r uE rAnBnr AEAB u rErDr u aEaDaDa u rEra r 5. 5. 0 111 0 1 11 1 1 1 22 11 11 1 12 2 1 1 2 21 21 2 1 2 2 ( , )cossin ( , )cossin=sin 0 ( , )sin . ( , )sinsin ( , )sin =0 nn nnnn n nn nnnn n n u rAnBnC r
38、D r u rAnBnC rD r A u rC rDrthis r C u rC rDr rr rr u rC rDr D rr 2 1 2 121 11 21 2 222222 122112 ( , )sinsin rrrr u rrrrr r rrrrrr 6. 6. 0 0 22 12 00 2 12 012 0 cossin ( , )cossin ( , )cossin( ) 111 ( )coscoscos0 111 22 1 ( )sin nnn n nn n nn n n nn n n n n n n AnBn RC r u rrAnBn u aaAnBnf a Afn du
39、n dun d uu a Au du d a Bfn d 22 12 12 00 1212 0 11 sinsin11 ( , )11sin 2 n n n n n uu un dun d n uuuu u rrn n a 9.1 2. 2. 22 2 22 2 2 4 4 xxi x i xi x F eeedx eedx e 9.2 6. 6. 2 2 22 2 22 2 2 2 2 2 2 2 2 1221 4 1 ,01 ,for ,01 1 11 1 2 txx txx at at atat x atat i x ua u u xx F uF a ux d F uia F u dt
40、d F uaF u dt F uCe F u xCF x F uF xe uFF xexFe e Feed 2 22 22 44 2 222 2 1 *121 22 a t xx a ta t at ee uxxdxa t atat 9.3 4. 4. 2 2 2 2 1112 112 222 3 02 23 23 1 23 23 111 21 3* 11 23*2 t t t t t t tttttttt yye y pL yL yL e L e L y p L e yLLLL e ppp LLe pp eeeeeeee 9.4 1. 1. 1 transform for 11 0,1 1 11 Laplacey d pL u dx x L uC p L uyCL y x L uL y p x uLyxyy p 10.1 1. 1. x f xxa dxa f a