數學物理方程(王明新等,清華大學出版社) 課后習題答案.pdf

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c Sut kSuxx k1Lu I2R a2 k c b k1L c S u0 I2R c S ut a2uxx bu u0 ut a2uxx bu u0 0 x 0 u 0 t u l t 0 t 0 u x 0 0 0 x l 1 11 a2 12 a11a12 a11uxx 2a12uxy a22uyy b1ux b2uy cu f 1 4 x x y y x y x y 1 4 a 11u 2a 12u a 22u b 1u b 2u c u f a 11 TA a 22 TA a 12 TA TA 8 A a11a12 a12a22 x y T x y T A a 11 a 12 a 12 a 22 TA TA TA TA JAJT J T Jacobi A A J 2 J 2 J 26 0 1 12 1 x2uxx y2uyy 0 2 uxx x y 2uyy 0 3 uxx xyuyy 0 4 xuxx 4uyy f x y 1 x2y2 xy 0 0 xy 6 0 0 2 x y 2 x y 0 0 x y 6 0 0 0 xy 0 4 4x x 0 0 x 0 0 x 0 1 13 1 uxx 4uxy 5uyy ux 2uy 0 2 uxx yuyy 0 3 uxx 4uxy 4uyy ey 4 uxx uxy uyy ux 0 1 1 dy2 4dxdy 5dx2 0 dy dx 2 i y 2x ix C 9 y 2x x u u u 0 2 y y 0 0 uxx 0 y 0 0 dy2 ydx2 0 2 y ix C 2 y x u u 1 u 0 y 0 dy2 ydx2 0 2 y x C x 2 y x 2 y u 1 2 u u 3 0 dy2 4dxdy 4dx2 0 y 2x C y 2x y 10 4u e 4 3 4 dy2 dxdy dx2 0 y 1 2 3 2 i x C x 2y x 3u u u u 0 1 14 1 uxx 3uxy 2uyy 0 2 uxx uxy 0 3 x2uxx 2xyuxy y2uyy xyux y2uy 0 1 1 4 0 dy2 3dxdy 2dx2 0 y x C1 y 2x C2 y x y 2x u 0 u f g u f x y g 2x y 2 1 4 0 dy2 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x t X x T t 3 X n Xn x 4 n T Tn t 5 X n Xn x Tn t 6 auxx buxy cuyy dux euy fu 0 2 1 a b c d e f u x y e xe y ux u uxx 2u uy u uyy 2u uxy u 2 1 u x y e xe y 2 1 a 2 b c 2 d e f 0 a 6 0 1 2 c 6 0 1 2 uxx uyy 0 ut uxx utt uxx 2 2 2 1 17 1 u u 0 0 x l u 0 u l 0 2 u u 0 0 x 2 u 0 u 2 u 0 u 2 3 u u 0 0 x l u 0 u l 0 1 0 u Z l 0 u x 2dx Z l 0 u2 x dx 0 0 2 0 u x C1cos x C2sin x u 0 0 C2 0 u l 0 C1cos l 0 u x cos l 0 2n 1 2l n 2n 1 2l 2 n 1 2 un x cos 2n 1 2l x 2 1 0 u x 0 0 2 0 u x C1cos x C2sin x C1 C1cos2 C2sin2 C2 C2cos2 C1sin2 cos2 1 n n 1 2 n n2 n 1 2 un x C1cosnx C2sinnx n n2 un 1 x sinnx un 2 x cosnx n 0 1 3 1 0 2 0 u x C1cos x C2sin x C2 0 C1 sin l 0 u C16 0 sin l 0 n l n 1 2 n n l 2 un x cos n x l n n l 2 un x cos n x l n 0 1 2 2 1 1 cosx cos2x cos3x 0 x 2 sin x sin2 x sin3 x 1 x 1 3 1 1 x 1 2x 1 2x 2 r x e x x 0 1 m n 0 Z 0 cosmxcosnxdx 2 n m 6 0 n m 0 0 n 6 m 0 2 m n 1 Z 1 1 sinm xsinn xdx 1 n m 0 n 6 m 3 u1 1 u2 1 x u3 1 2x 1 2x 2 Z 0 u2 ie xdx 0 i 1 2 3 Z 0 u1u2e xdx 1 2 0 Z 0 u1u3e xdx 1 2 2 1 2 3 0 Z 0 u2u3e xdx 1 3 2 5 2 3 1 2 4 0 r x e x 2 3 0 l d dx k x dy dx q x y x x y x 0 y 0 y l 0 19 k x k0 0 q x q0 0 x 0 0 0 x y 0 l ky dy dx l 0 Z l 0 k dy dx 2 dx Z l 0 qy2dx Z l 0 y2dx 0 ky dy dx l 0 0 Z l 0 y2dx Z l 0 k dy dx 2 dx Z l 0 qy2dx 0 0 x 1 2 y1 y2 d dx k x dy1 dx qy1 1 y1 0 2 2 d dx k x dy2 dx qy2 2 y2 0 2 3 2 2 y2 2 3 y1 0 l k dy1 dx y2 l 0 Z l 0 k dy1 dx dy2 dx dx Z l 0 qy1y2dx 1 Z l 0 y1y2dx 0 k dy2 dx y1 l 0 Z l 0 k dy1 dx dy2 dx dx Z l 0 qy1y2dx 2 Z l 0 y1y2dx 0 k dy1 dx y2 l 0 k dy2 dx y1 l 0 0 Z l 0 y1y2dx 0 2 4 1 utt uxx 0 x 0 u 0 t u 2 t 0 t 0 u x 0 1 2 sin x ut x 0 0 0 x 2 2 utt a2uxx 0 0 x 0 ux 0 t u l t 0 t 0 u x 0 cos 2l x 0 x l ut x 0 cos 3 2l x cos 5 2l x 0 x l 3 utt a2uxx 0 x 0 u 0 t u 1 t 0 t 0 u x 0 x ut x 0 x 0 x 1 20 x x 1 x x x 0 x 1 2 1 x 1 2 x 1 4 utt a2uxx 0 x 0 u 0 t ux t 0 t 0 u x 0 x3 ut x 0 0 0 x 1 u x t X x T t T t T t X x X x X X 0 T T 0 u 0 t u 2 t 0 X 0 X 2 0 X X 0 0 x 2 X 0 X 2 0 n n 2 2 Xn x sin n x 2 n T Tn t Cncos n t 2 Dnsin n t 2 n 1 2 u x t X n 1 sin n x 2 Cncos n t 2 Dnsin n t 2 u x 0 X n 1 Cnsin n x 2 1 2 sin x ut x 0 X n 1 n 2 Dnsin n x 2 0 Dn 0 Cn 1 2 n 2 0 n 6 2 u x t 1 2 cos tsin x 2 u x t X x T t T t a2T t X x X x X X 0 T a2T 0 21 ux 0 t u l t 0 X 0 X l 0 X X 0 0 x l X 0 X l 0 n 2n 1 2l 2 Xn x cos 2n 1 x 2l n 1 2 n T Tn t Cncos 2n 1 at 2l Dnsin 2n 1 at 2l n 1 2 u x t X n 1 cos 2n 1 x 2l Cncos 2n 1 at 2l Dnsin 2n 1 at 2l Cn 2 l Z l 0 cos 2n 1 x 2l cos x 2l dx 1 n 1 0 n 6 1 Dn 4 2n 1 a Z l 0 cos 2n 1 x 2l cos 3 2l x cos 5 2l x dx 2l 3 a n 2 2l 5 a n 3 0 n 1 4 5 u x t cos at 2l cos x 2l 2l 3 a sin 3 at 2l cos 3 x 2l 2l 5 a sin 5 at 2l cos 5 x 2l 3 1 u x t X n 1 sinn x Cncosn at Dnsinn at Cn 2 Z 1 2 0 xsinn xdx Z 1 1 2 1 x sinn xdx 4 n 2 sin n 2 Dn 2 n a Z 1 0 x 1 x sinn xdx 4 a n 4 1 1 n u x t X n 1 4 n 2 sin n 2 cosn at 4 a n 4 1 1 n sinn at sinn x 22 4 2 u x t X n 1 sin 2n 1 x 2 Cncos 2n 1 at 2 Dnsin 2n 1 at 2 Cn 2 Z 0 x3sin 2n 1 x 2 dx 24 1 n 1 2n 1 2 2 8 2n 1 2 Dn 0 u x t X n 1 24 1 n 1 2n 1 2 2 8 2n 1 2 cos 2n 1 at 2 sin 2n 1 x 2 2 5 l x ut a2uxx 0 x 0 ux 0 t ux l t 0 t 0 u x 0 x 0 x l u x t X x T t T t a2T t X x X x ux 0 t ux l t 0 X 0 X l 0 X X 0 0 x l X 0 X l 0 n n l 2 Xn x cos n x l n 0 1 n T Tn t Cnexp n a l 2 t n 0 1 2 u x t X n 0 Cnexp n a l 2 t cos n x l Cn 2 l Z l 0 xcos n x l dx 2l n2 2 1 n 1 n 1 2 1 l Z l 0 xdx l 2 n 0 u x t l 2 X n 1 2l n2 2 1 n 1 exp n a l 2 t cos n x l l 2 X n 1 4l 2n 1 2 2 exp 2n 1 a l 2 t cos 2n 1 x l 23 2 6 1 ut a2uxx 0 x 0 u 0 t u l t 0 t 0 u x 0 x l x 0 x l 2 ut a2uxx 0 x 0 ux 0 t ux t 0 t 0 u x 0 sinx 0 x 3 ut a2uxx 0 0 x 0 ux 0 t u 0 t 0 t 0 ux l t u l t 0 t 0 u x 0 x 0 x l 0 1 2 5 u x t X n 1 Cnexp n a l 2 t sin n x l Cn 2 l Z l 0 x l x sin n x l dx 4l2 n 3 1 1 n u x t X n 1 4l2 n3 3 1 1 n exp n a l 2 t sin n x l X n 1 8l2 2n 1 3 3 exp 2n 1 a l 2 t sin 2n 1 x l 2 2 5 u x t X n 0 Cne n 2a2t cosnx C0 1 Z 0 sinxdx 2 Cn 2 Z 0 cosnxsinxdx 0 n 1 2 1 n 1 1 n2 1 n 2 u x t 2 2 X n 2 1 1 n n2 1 e n 2a2t cosnx 3 u x t X x T t T T a2 T 0 X X x X 0 0 x l X 0 X 0 0 X l X l 0 2 4 24 2 4 X x Acos x B sin x B A 0 tan l B A A B 2 2 tan l 2 2 0 n n 2 4 Xn x Ancos p nx Bnsin p nx Bn sin p nx n cos p nx Knsin p nx n n arctan n Kn Bn p 2 n n T Tn t Cne a 2 nt u x t X n 1 Dne a 2 nt sin p nx n Dn Z l 0 sin p nx n x dx Z l 0 sin2 p nx n dx 2 7 l x 0 u0 x l u x t ut a2uxx b2u 0 x 0 u 0 t u0 ux u x l 0 t 0 u x 0 0 0 x l u x t u x t e b 2tv x t w x v x t vt a2vxx 0 0 x 0 v 0 t vx v x l 0 t 0 v x 0 w x 0 x l 2 5 w x a2w x b2w x 0 0 x l w 0 u0 w l w l 0 2 6 2 6 w x Ae x Be x b a A B u0 Ae l Be l Ae l Be l 0 25 A u0 e l 2 cosh l sinh l B u0 e l 2 cosh l sinh l w x cosh x l sinh x l cosh l sinh l u0 v x t 2 5 v x t X x T t T T a2 T 0 X X x X 0 0 x l X 0 X l X l 0 tan l 0 n n Xn x sin nx n T Tn t Cne a 2 nt v x t X n 1 Cne a 2 nt sin p nx Cn Z l 0 w x sin p nxdx Z l 0 sin2 p nxdx 2 8 Dirichlet uxx uyy 0 x2 y2 a2 u x2 y2 a2 f f 1 f A 2 f Acos 3 f Asin2 B cos2 Dirichlet u A0 2 X n 1 n Ancosn Bnsinn An 1 an Z 2 0 f cosn d n 0 1 2 Bn 1 an Z 2 0 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r 1 r 1 r 1 r 1 4i ln 1 r2 r 1 r2 r 1 4i ln 1 r2 2irsin 1 r2 2irsin 1 2 arctan 2rsin 1 r2 2 10 1 uxx uyy 0 0 x 0 y ux 0 y y 2 ux y 0 0 y uy x 0 uy x 0 0 x 2 uxx uyy 0 0 x a 0 y b u 0 y u a y 0 0 y b u x 0 0 u x b f x 0 x a 3 uxx uyy 0 0 x a 0 y b u 0 y 0 u a y Ay 0 y b uy x 0 uy x b 0 0 x a 1 u x y X x Y y X X Y Y X X 0 Y Y 0 Y 0 Y 0 Y Y 0 0 y Y 0 Y 0 n n2 Yn y cosny n 0 1 2 X nX 0 X0 x a0 a1x n 0 Xn x Cnenx Dne nx n 1 2 29 u x y a0 a1x X n 1 Cnenx Dne nx cosny ux 0 y y 2 a1 X n 1 n Cn Dn cosny ux y 0 a1 X n 1 n Cnen Dne n cosny a1 0 Cnen Dne n 0 Cn Dn 2 n3 1 n 1 Cn e n 1 1 n n3 sinhn Dn en 1 1 n n3 sinhn u x y a0 X n 1 2 1 1 n n3 coshn x sinhn cosny a0 2 u x y X x Y y X X Y Y X X 0 Y Y 0 X 0 X a 0 X X 0 0 x a X 0 X a 0 n n a 2 Xn x sin n x a n 1 2 Y nY 0 Yn y Cnen y a Dne n y a u x y X n 1 C nen y a Dne n y a sin n x a 30 Cn Dn 0 Cnen b a Dne n b a 2 a Z a 0 f sin n a d Cn Z a 0 f sin n a d asinh n b a Dn Z a 0 f sin n a d asinh n b a u x y 2 a X n 1 Z a 0 f sin n a d sinhn y a sinh n b a sin n x a 3 u x y X x Y y X X Y Y X X 0 Y Y 0 Y 0 Y b 0 Y Y 0 0 y b Y 0 Y b 0 n n b 2 Yn y cos n y b n 0 1 X nX 0 X0 x C0 x D0 n 0 Xn x Cnen x b Dne n x b n 1 2 u x y C0 x D0 X n 1 C nen x b Dne n x b cos n y b u 0 y 0 D0 X n 1 Cn Dn cos n y b u a y Ay C0a D0 X n 1 C nen a b Dne n a b cos n y b D0 0 C0 1 ab Z b 0 Aydy Ab 2a Cn Dn 0 Cnen a b Dne n a b 2Ab n 2 1 n 1 31 Cn Ab 1 n 1 n 2sinh n a b Dn Ab 1 1 n n 2sinh n a b u x t Abx 2a 2Ab 2 X n 1 1 n 1 n2 sinh n x b sinh n a b cos n y b Abx 2a 4Ab 2 X n 1 sinh 2n 1 x b sinh 2n 1 a b cos 2n 1 y b 2n 1 2 2 11 u urr 1 r ur 1 r2 u f r r1 r r2 u r1 u r2 0 0 2 f r f r u r u r rur r rf r rur Z r r1 f d C r u r Z r r1 1 t dt Z t r1 f d C lnr D Z r r1 f d Z r 1 t dt C lnr D Z r r1 f ln r d C lnr D C lnr1 D 0 C lnr2 D Z r2 r1 f ln r2 d C 1 lnr2 lnr1 Z r2 r1 f ln r2 d D lnr1 lnr2 lnr1 Z r2 r1 f ln r2 d 32 u r Z r r1 f ln r d lnr1 lnr lnr2 lnr1 Z r2 r1 f ln r2 d r1 r r2 2 12 uxx uyy f x y 0 x a 0 y b u 0 y 1 y u a y 2 y 0 y b u x 0 1 x u x b 2 x 0 x a u x y v x y 1 y x a 2 y 1 y v vxx vyy f x y 1 y x a 2 y 1 y 0 x a 0 y b v 0 y v a y 0 0 y b v x 0 1 x 1 0 x a 2 0 1 0 0 x a v x b 2 x 1 b x a 2 b 1 b 0 x a x v x y X n 1 fn y sin n x a fn y X n 1 f n y n a 2 fn y sin n x a f x y 1 y x a 2 y 1 y fn y f n y n a 2 fn y 2 a Z a 0 f x y 1 y x a 2 y 1 y sin n x a dx 0 y b 2 8 v x 0 1 x 1 0 x a 2 0 1 0 X n 1 fn 0 sin n x a v x b 2 x 1 b x a 2 b 1 b X n 1 fn b sin n x a fn 0 2 a Z a 0 1 x 1 0 x a 2 0 1 0 sin n x a dx fn b 2 a Z a 0 2 x 1 b x a 2 b 1 b sin n x a dx 2 9 fn y 2 8 2 9 u x y w x y 1 x y b 2 x 1 x 33 w y 2 13 a b 2 3 2 mn m2 a2 n2 b2 2 m n 1 2 T T t mnT t 0 Tmn t Cmnsin p mnt Dmncos p mnt m n 1 2 u x y t X m n 1 C mnsin p mnt Dmncos p mnt sin m x a sin n y b 2 10 u x y 0 x y X m n 1 Dmnsin m x a sin n y b ut x y 0 x y X m n 1 p mnCmnsin m x a sin n y b Cmn 4 ab mn Z b 0 Z a 0 x y sin m x a sin n y b dxdy Dmn 4 ab Z b 0 Z a 0 x y sin m x a sin n y b dxdy Cmn Dmn 2 10 2 14 1 utt a2uxx sin 2 x l sin 2 at l 0 x 0 u 0 t u l t 0 t 0 u x 0 ut x 0 0 0 x l 2 utt a2uxx 0 x 0 u 0 t t2 ux l t 0 t 0 u x 0 ut x 0 0 0 x l 3 utt a2uxx Asinhx 0 x 0 u 0 t k0 u l t k t 0 u x 0 ut x 0 0 0 x l A k0 k 4 ut a2uxx 0 x 0 u 0 t 0 u l t At t 0 u x 0 0 0 x l 34 A 1 u x t X n 1 2 n a sin n x l Z t 0 Z l 0 sin 2 l sin 2 a l sin n l sin n a l t d d 1 a sin 2 x l Z l 0 sin 2 l 2 d Z t 0 sin 2 a l sin 2 a l t d l 4 a l 2 a sin 2 at l tcos 2 at l sin 2 x l 2 u x t v x t t2 w x vtt a2vxx a2w x 2 0 x 0 v 0 t w 0 vx l t w l t 0 v x 0 w x vt x 0 0 0 x l v w a2w x 2 0 0 x l w 0 w l 0 w x x2 a2 2lx a2 w x v x t vtt a2vxx 0 0 x 0 v 0 t vx l t 0 t 0 v x 0 w x vt x 0 0 0 x l v x t X n 1 Cncos 2n 1 at 2l Dnsin 2n 1 at 2l sin 2n 1 x 2l Cn 2 l Z l 0 2lx a2 x2 a2 sin 2n 1 x 2l dx 32l2 2n 1 3 3a2 Dn 0 n 1 2 v x t X n 1 32l2 2n 1 3 3a2 sin 2n 1 x 2l cos 2n 1 at 2l u x t t2 x2 a2 2lx a2 X n 1 32l2 2n 1 3 3a2 sin 2n 1 x 2l cos 2n 1 at 2l 3 u x t v x t w x vtt a2vxx a2w x Asinhx 0 x 0 v 0 t k0 w 0 v l t k w l t 0 v x 0 w x vt x 0 0 0 x l 35 v w a2w x Asinhx 0 0 x l w 0 k0 w l k w x k0 A a2 sinhx A a2 sinhl k k0 x l w x v x t vtt a2vxx 0 0 x 0 v 0 t v l t 0 t 0 v x 0 w x vt x 0 0 0 x l v x t X n 1 Cncos n at l Dnsin n at l sin n x l Cn 2 l Z l 0 w sin n l d Dn 0 n 1 2 v x t 2 l X n 1 Z l 0 w sin n l d cos n at l sin n x l u x t w x 2 l X n 1 Z l 0 w sin n l d cos n at l sin n x l k0 A a2 sinhx A a2 sinhl k k0 x l 2 l X n 1 Z l 0 w sin n l d cos n at l sin n x l 4 u x t v x t A l xt w x vt a2vxx a2w x A l x 0 x 0 v 0 t w 0 v l t w l t 0 v x 0 w x 0 x l v w a2w x A l x 0 0 x l w 0 w l 0 w x A 6a2l x x2 l2 36 w x v x t vt a2vxx 0 0 x 0 v 0 t v l t 0 t 0 v x 0 A 6a2l x x2 l2 0 x l v x t X n 1 Cnexp n a l 2 t sin n x l Cn 2 l Z l 0 A 6a2l x x2 l2 sin n x l dx 1 n 12Al2 n3 3a2 v x t X n 1 1 n 1 2Al2 n3 3a2 exp n a l 2 t sin n x l u x t A 6a2l x x2 l2 A l xt X n 1 1 n 1 2Al2 n3 3a2 exp n a l 2 t sin n x l 2 15 l A 1 u x t ut a2uxx 0 x 0 ux 0 t ux l t t 0 u x 0 A 0 x l u x t v x t l x2 x vt a2vxx 2a 2 l 0 x 0 vx 0 t vx l t 0 t 0 v x 0 A l x2 x 0 x l v x t w x t A 2a2 l t w wt a2wxx 0 0 x 0 wx 0 t wx l t 0 t 0 w x 0 l x2 x 0 x l w x t C0 X n 1 Cnexp n a l 2 t cos n x l 37 C0 1 l Z l 0 l x2 x dx l 6 n 0 Cn 2 l Z l 0 l x2 x cos n x l dx 2 l n2 2 1 1 n n 1 2 w x t l 6 X n 1 2 l n2 2 1 1 n exp n a l 2 t cos n x l l 6 X n 1 l n2 2 exp 2n a l 2 t cos 2n x l u x t l x2 x A 2a2 l t l 6 X n 1 l n2 2 exp 2n a l 2 t cos 2n x l 2 16 1 utt a2uxx 2x l sin t 0 x 0 u 0 t t u l t sin t t 0 u x 0 0 ut x 0 0 x l 2 utt a2uxx bux 0 x 0 u 0 t 0 u l t Bt t 0 u x 0 ut x 0 0 0 x l 3 utt uxx ex 0 x 0 ux 0 t 1 u l t t t 0 u x 0 ut x 0 0 0 x l 4 ut a2uxx b2u 0 x 0 u 0 t u l t 0 t 0 u x 0 x 0 x l b B x 1 u x t v x t t x l sin t t v vtt a2vxx 0 0 x 0 v 0 t v l t 0 t 0 v x 0 vt x 0 0 0 x l v x t 0 u x t t x l sin t t 38 2 u e bx 2a 2 v v vtt a2vxx b2 4a2 v 0 x 0 v 0 t 0 v l t ebl 2a 2 Bt t 0 v x 0 vt x 0 0 0 x l v x t w x t B l ebl 2a 2 xt w wtt a2wxx b2 4a2 w b2B 4a2l ebl 2a 2 xt 0 x 0 w 0 t w l t 0 t 0 w x 0 0 wt x 0 B l ebl 2a 2 x 0 x l w x t p x t q x t