Типовые

11

Задача 9. Решить смешанную задачу.

9.1.

ut=

1

uxx + 5cos2t sin2x ;

u(x,0)=sin4x ;

u(0,t)=u(π,t)=0

4

9.2.

ut=

1

uxx + 5sin2t sin3x ;

u(x,0)=2sin9x ;

u(0,t)=u(π,t)=0

9

9.3.

ut=

1

uxx + 10cos3t sin4x ;

u(x,0)=3sin16x ;

u(0,t)=u(π,t)=0

16

9.4.

ut=

1

uxx + 10sin3t sin5x ;

u(x,0)=4sin10x ;

u(0,t)=u(π,t)=0

25

9.5.

ut=

1

uxx + 17cos4t sin6x ;

u(x,0)=5sin18x ;

u(0,t)=u(π,t)=0

36

9.6.

ut=

1

uxx + 17sin4t sin2x ;

u(x,0)=6sin8x ;

u(0,t)=u(π,t)=0

4

9.7.

ut=

1

uxx + 26cos5t sin3x ;

u(x,0)=7sin6x ;

u(0,t)=u(π,t)=0

9

9.8.

ut=

1

uxx + 26sin5t sin4x ;

u(x,0)=8sin12x ;

u(0,t)=u(π,t)=0

16

9.9.

ut=

1

uxx + 37cos6t sin5x ;

u(x,0)=9sin20x ;

u(0,t)=u(π,t)=0

25

9.10.

ut=

1

uxx + 37sin6t sin6x ;

u(x,0)=10sin12x ;

u(0,t)=u(π,t)=0

36

9.11.

ut=

1

uxx + 26cos5t sin2x ;

u(x,0)=11sin6x ;

u(0,t)=u(π,t)=0

4

9.12.

ut=

1

uxx + 26sin5t sin3x ;

u(x,0)=12sin12x ;

u(0,t)=u(π,t)=0

9

9.13.

ut=

1

uxx + 17cos4t sin4x ;

u(x,0)=13sin8x ;

u(0,t)=u(π,t)=0

16

9.14.

ut=

1

uxx + 17sin4t sin5x ;

u(x,0)=14sin20x ;

u(0,t)=u(π,t)=0

25

9.15.

ut=

1

uxx + 10cos3t sin6x ;

u(x,0)=15sin15x ;

u(0,t)=u(π,t)=0

36

12

9.16.

ut=

1

uxx + 10sin3t sin2x ;

u(x,0)=16sin18x ;

u(0,t)=u(π,t)=0

4

9.17. ut=

1

uxx + 5cos2t sin3x ;

u(x,0)=17sin4x ;

u(0,t)=u(π,t)=0

9

9.18.

ut=

1

uxx + 5sin2t sin4x ;

u(x,0)=18sin9x ;

u(0,t)=u(π,t)=0

16

9.19.

ut=

1

uxx + 10cos3t sin5x ;

u(x,0)=19sin16x ;

u(0,t)=u(π,t)=0

25

9.20.

ut=

1

uxx + 10sin3t sin6x ;

u(x,0)=20sin10x ;

u(0,t)=u(π,t)=0

36

9.21.

ut=

1

uxx + 17cos4t sin2x ;

u(x,0)=21sin18x ;

u(0,t)=u(π,t)=0

4

9.22. ut=

1

uxx + 17sin4t sin3x ;

u(x,0)=22sin8x ;

u(0,t)=u(π,t)=0

9

9.23.

ut=

1

uxx + 26cos5t sin4x ;

u(x,0)=23sin6x ;

u(0,t)=u(π,t)=0

16

9.24.

ut=

1

uxx + 26sin5t sin5x ;

u(x,0)=24sin12x ;

u(0,t)=u(π,t)=0

25

9.25.

ut=

1

uxx + 37cos6t sin6x ;

u(x,0)=25sin20x ;

u(0,t)=u(π,t)=0

36

9.26.

ut=

1

uxx + 37sin6t sin2x ;

u(x,0)=26sin12x ;

u(0,t)=u(π,t)=0

4

9.27.

ut=

1

uxx + 26cos5t sin3x ;

u(x,0)=27sin6x ;

u(0,t)=u(π,t)=0

16

9.28. ut=

1

uxx + 26sin5t sin4x ;

u(x,0)=28sin12x ;

u(0,t)=u(π,t)=0

9

9.29.

ut=

1

uxx + 17cos4t sin6x;

u(x,0)=29sin8x ;

u(0,t)=u(π,t)=0

25

9.30.

ut=

1

uxx +17sin4t sin5x ;

u(x,0)=30sin18x ;

u(0,t)=u(π,t)=0

36

9.31.

ut=

1

uxx + 10cos3t sin4x ;

u(x,0)=31sin20x ;

u(0,t)=u(π,t)=09

4

10.1.

ut =

1

uxx + 5sin2t sin6x;

u(x,0)=sin12x + π + 3x;

u(0,t)=π,

u(π,t)=4π

36

10.2.

ut =

1

uxx + 5cos2t sin2x;

u(x,0)=2sin6x − π + 2x;

u(0,t)=−π,

u(π,t)=π

4

10.3.

ut =

1

uxx + 10sin3t sin3x;

u(x,0)=3sin12x + 2π − x;

u(0,t)= 2π,

u(π,t)=π

9

10.4.

ut =

1

uxx + 10cos3t sin4x;

u(x,0)=4sin8x − 2π + x;

u(0,t)= −2π,

u(π,t)=−π

16

10.5.

ut =

1

uxx + 17sin4t sin5x;

u(x,0)=5sin15x + 3π − 2x;

u(0,t)=3π,

u(π,t)=π

25

10.6.

ut =

1

uxx + 17cos4t sin6x;

u(x,0)=6sin24x − 4π + 2x;

u(0,t)=−4π,

u(π,t)=−2π

36

10.7.

ut =

1

uxx + 26sin5t sin2x;

u(x,0)=7sin8x + 4π − 3x;

u(0,t)= 4π,

u(π,t)=π

4

10.8.

ut =

1

uxx + 26cos5t sin3x;

u(x,0)=8sin9x − 3π + 3x;

u(0,t)= −3π,

u(π,t)=0

9

10.9.

ut =

1

uxx + 37sin6t sin4x;

u(x,0)=9sin8x + 5π + 4x;

u(0,t)=5π,

u(π,t)=π

16

10.10. ut =

1

uxx + 37cos6t sin5x;

u(x,0)=10sin10x − 5π − 4x;

u(0,t)=−5π,

u(π,t)=−π

25

10.11. ut =

1

uxx + 26sin5t sin6x;

u(x,0)=11sin18x + π − 2x;

u(0,t)= 2π,

u(π,t)=−π

36

10.12. ut =

1

uxx + 26cos5t sin2x;

u(x,0)=12sin4x + 2π − 3x;

u(0,t)=π,

u(π,t)=−π

4

10.13. ut =

1

uxx + 17sin4t sin3x;

u(x,0)=13sin6x + 3π − 4x;

u(0,t)= 3π,

u(π,t)=−π

9

10.14. ut =

1

uxx + 17cos4t sin4x;

u(x,0)=14sin8x + 4π − 5x;

u(0,t)= 4π,

u(π,t)=−π

16

10.15. ut =

1

uxx + 10sin3t sin5x;

u(x,0)=15sin15x + 5π − 6x;

u(0,t)= 5π,

u(π,t)=−π

25

10.16. ut =

1

uxx + 10cos3t sin3x;

u(x,0)=16sin6x

− 5π + 6x;

u(0,t)= −5π,

u(π,t)=π

36

10.17. ut =

1

uxx + 5sin2t sin2x;

u(x,0)=17sin6x

− 4π + 5x;

u(0,t)= −4π,

u(π,t)=π

4

10.18. ut =

1

uxx + 5cos2t sin3x;

u(x,0)=18sin9x

− 3π + 4x;

u(0,t)= −3π,

u(π,t)=π

9

10.19. ut =

1

uxx + 10sin3t sin4x;

u(x,0)=19sin15x − 2π + 3x;

u(0,t)= −2π,

u(π,t)=π

16

10.20. ut =

1

uxx + 10cos3t sin5x;

u(x,0)=20sin10x

− π + 2x;

u(0,t)=−π,

u(π,t)=π

25

10.21. ut =

1

uxx + 17sin4t sin6x;

u(x,0)=21sin12x + 5π − 3x;

u(0,t)= 5π,

u(π,t)=2π

36

10.22. ut =

1

uxx + 17cos4t sin2x;

u(x,0)=22sin8x − 4π + 4x;

u(0,t)= −5π,

u(π,t)=−π

4

10.23. ut =

1

uxx + 26sin5t sin3x;

u(x,0)=23sin12x + 4π − 4x;

u(0,t)= 4π,

u(π,t)=0

9

10.24. ut =

1

uxx + 26cos5t sin4x;

u(x,0)=24sin16x

− 4π + 5x;

u(0,t)= −4π,

u(π,t)=π

16

10.25. ut =

1

uxx + 37sin6t sin5x;

u(x,0)=25sin15x + 3π − 5x;

u(0,t)= 3π,

u(π,t)=−2π

25

10.26. ut =

1

uxx + 37cos6t sin6x;

u(x,0)=26sin18x

− 3π + 5x;

u(0,t)= −3π,

u(π,t)=3π

36

10.27. ut =

1

uxx + 26sin5t sin2x;

u(x,0)=27sin10x + 2π − 6x;

u(0,t)= 2π,

u(π,t)=−4π

4

10.28. ut =

1

uxx + 26cos5t sin3x;

u(x,0)=28sin15x

− 2π + 2x;

u(0,t)= −2π,

u(π,t)=0

9

10.29. ut =

1

uxx + 17sin4t sin4x;

u(x,0)=29sin20x + π − x;

u(0,t)=π,

u(π,t)=0

16

10.30. ut =

1

uxx + 17cos4t sin5x;

u(x,0)=20sin20x

− π + x;

u(0,t)=−π,

u(π,t)=0

25

10.31. ut =

1

uxx + 10sin3t sin6x;

u(x,0)=31sin24x + π + x;

u(0,t)=π,

u(π,t)=2π

36

11.1.

u(1,ϕ)=sin6ϕ;

u (r,0)=u (r,π/3)=0

11.2.

u(1,ϕ)=2cos2ϕ;

uϕ(r,0)=uϕ(r,π)=0

11.3.

u(1,ϕ)=3cos15ϕ;

uϕ(r,0)=u (r,π/6)=0

11.4.

u(1,ϕ)=4sin14ϕ;

u (r,0)=uϕ (r,π/4)=0

11.5.

u(1,ϕ)=5sin3ϕ;

u (r,0)=u (r,2π/3)=0

11.6.

u(1,ϕ)=6cos6ϕ;

uϕ(r,0)=uϕ(r,7π/6)=0

11.7.

u(1,ϕ)=7cos10ϕ;

uϕ(r,0)=u (r,π/4)=0

11.8.

u(1,ϕ)=8sin7ϕ;

u (r,0)=uϕ (r,π/2)=0

11.9.

u(1,ϕ)=9sin4ϕ;

u (r,0)=u (r,3π/4)=0

11.10.

u(1,ϕ)=10cos4ϕ;

uϕ(r,0)=uϕ(r,5π/4)=0

11.11.

u(1,ϕ)=11cos5ϕ;

uϕ(r,0)=u (r,π/2)=0

11.12.

u(1,ϕ)=12sin3ϕ;

u (r,0)=uϕ (r,3π/2)=0

11.13.

u(1,ϕ)=13sin6ϕ;

u (r,0)=u (r,5π/6)=0

11.14.

u(1,ϕ)=14cos3ϕ;

uϕ(r,0)=uϕ(r,4π/3)=0

11.15.

u(1,ϕ)=15cosϕ;

uϕ(r,0)=u (r,3π/2)=0

11.16.

u(1,ϕ)=16sin21ϕ;

u (r,0)=uϕ (r,π/6)=0

11.17.

u(1,ϕ)=17sin9ϕ;

u (r,0)=u (r,π/3)=0

11.18.

u(1,ϕ)=18cos4ϕ;

uϕ(r,0)=uϕ(r,π)=0

11.19.

u(1,ϕ)=19cos21ϕ;

uϕ(r,0)=u (r,π/6)=0

11.20.

u(1,ϕ)=20sin15ϕ;

u (r,0)=uϕ (r,π/6)=0

11.21.

u(1,ϕ)=21sin6ϕ;

u (r,0)=u (r,2π/3)=0

11.22.

u(1,ϕ)=22cos12ϕ;

uϕ(r,0)=uϕ(r,π/3)=0

11.23.

u(1,ϕ)=23cos14ϕ;

uϕ(r,0)=u (r,π/4)=0

11.24.

u(1,ϕ)=24sin10ϕ;

u (r,0)=uϕ (r,π/4)=0

11.25.

u(1,ϕ)=25sin3ϕ;

u (r,0)=u (r,π)=0

11.26.

u(1,ϕ)=26cos3ϕ;

uϕ(r,0)=uϕ(r,5π/3)=0

11.27.

u(1,ϕ)=27cos7ϕ;

uϕ(r,0)=u (r,π/2)=0

11.28.

u(1,ϕ)=28sin5ϕ;

u (r,0)=uϕ (r,π/2)=0

11.29.

u(1,ϕ)=29sin3ϕ;

u (r,0)=u (r,5π/3)=0

11.30.

u(1,ϕ)=20cos4ϕ;

uϕ(r,0)=uϕ(r,7π/4)=0

11.31.

u(1,ϕ)=31cos3ϕ;

uϕ(r,0)=u (r,3π/2)=0.

12.1.

u(1,ϕ)=cos9ϕ;

12.2.

u(1,ϕ)=2sin8ϕ;

12.3.

u(1,ϕ)=3cos7ϕ;

12.4.

u(1,ϕ)=4sin6ϕ;

12.5.

u(1,ϕ)=5cos5ϕ;

12.6.

u(1,ϕ)=6sin4ϕ;

12.7.

u(1,ϕ)=7cos3ϕ;

12.8.

u(1,ϕ)=8sin2ϕ;

12.9.

u(1,ϕ)=9cos2ϕ;

12.10. u(1,ϕ)=10sin3ϕ;

12.11. u(1,ϕ)=11cos4ϕ;

12.12. u(1,ϕ)=12sin5ϕ;

12.13. u(1,ϕ)=13cos6ϕ;

12.14. u(1,ϕ)=14sin7ϕ;

12.15. u(1,ϕ)=15cos8ϕ;

12.16. u(1,ϕ)=16sin9ϕ;

12.17. u(1,ϕ)=17cos9ϕ;

12.18. u(1,ϕ)=18sin8ϕ;

12.19. u(1,ϕ)=19cos7ϕ;

12.20. u(1,ϕ)=20sin6ϕ;

12.21. u(1,ϕ)=21cos5ϕ;

12.22. u(1,ϕ)=22sin4ϕ;

12.23. u(1,ϕ)=23cos3ϕ;

12.24. u(1,ϕ)=24sin2ϕ;

12.25. u(1,ϕ)=25cos2ϕ;

12.26. u(1,ϕ)=26sin3ϕ;

12.27. u(1,ϕ)=27cos4ϕ;

12.28. u(1,ϕ)=28sin5ϕ;

12.29. u(1,ϕ)=29cos6ϕ;

12.30. u(1,ϕ)=30sin7ϕ;

12.31. u(1,ϕ)=31cos8ϕ+32sin9.

17

Задача 13. Решить смешанную задачу.

13.1.

utt =81uxx

u(x,0)=sinπx;

ut(x,0)=0;

u(0,t)=u(5,t)=0

13.2.

utt =64uxx

u(x,0)=0;

ut(x,0)= 8πsinπx;

u(0,t)=u(6,t)=0

13.3.

utt =49uxx

u(x,0)=3sin2πx;

ut(x,0)=0;

u(0,t)=u(4,t)=0

13.4.

utt =36uxx

u(x,0)=0;

ut(x,0)= 12πsin2πx;

u(0,t)=u(5,t)=0

13.5.

utt =25uxx

u(x,0)=5sin3πx;

ut(x,0)=0;

u(0,t)=u(3,t)=0

13.6.

utt =16uxx

u(x,0)=0;

ut(x,0)= 12πsin3πx;

u(0,t)=u(4,t)=0

13.7.

utt =9uxx

u(x,0)=7sin4πx;

ut(x,0)=0;

u(0,t)=u(2,t)=0

13.8.

utt =4uxx

u(x,0)=0;

ut(x,0)= 8πsin4πx;

u(0,t)=u(3,t)=0

13.9.

utt =uxx

u(x,0)=9sin5πx;

ut(x,0)=0;

u(0,t)=u(1,t)=0

13.10.

utt =uxx

u(x,0)=0;

ut(x,0)= 5πsin5πx;

u(0,t)=u(3,t)=0

13.11.

utt =4uxx

u(x,0)=11sin6πx;

ut(x,0)=0;

u(0,t)=u(2,t)=0

13.12.

utt =9uxx

u(x,0)=0;

ut(x,0)= 18πsin6πx;

u(0,t)=u(1,t)=0

13.13.

utt =19uxx

u(x,0)=13sin5πx;

ut(x,0)=0;

u(0,t)=u(3,t)=0

13.14.

utt =25uxx

u(x,0)=0;

ut(x,0)= 25πsin5πx;

u(0,t)=u(2,t)=0

13.15.

utt =36uxx

u(x,0)=15sin4πx;

ut(x,0)=0;

u(0,t)=u(4,t)=0

13.16.

utt =49uxx

u(x,0)=0;

ut(x,0)= 28πsin4πx;

u(0,t)=u(3,t)=0

13.17.

utt =64uxx

u(x,0)=17sin3πx;

ut(x,0)=0;

u(0,t)=u(5,t)=0

13.18.

utt =81uxx

u(x,0)=0;

ut(x,0)= 27πsin3πx;

u(0,t)=u(4,t)=0

13.19.

utt =uxx

u(x,0)=19sin7πx;

ut(x,0)=0;

u(0,t)=u(2,t)=0

13.20.

utt =4uxx

u(x,0)=0;

ut(x,0)= 14πsin7πx;

u(0,t)=u(1,t)=0

13.21.

utt =9uxx

u(x,0)=21sin6πx;

ut(x,0)=0;

u(0,t)=u(3,t)=0

13.22.

utt =16uxx

u(x,0)=0;

ut(x,0)= 24πsin6πx;

u(0,t)=u(2,t)=0

13.23.

utt =25uxx

u(x,0)=23sin5πx;

ut(x,0)=0;

u(0,t)=u(4,t)=0

13.24.

utt =36uxx

u(x,0)=0;

ut(x,0)= 30πsin5πx;

u(0,t)=u(1,t)=0

13.25.

utt =49uxx

u(x,0)=25sin4πx;

ut(x,0)=0;

u(0,t)=u(5,t)=0

13.26.

utt =64uxx

u(x,0)=0;

ut(x,0)= 32πsin4πx;

u(0,t)=u(4,t)=0

13.27.

utt =81uxx

u(x,0)=27sin3πx;

ut(x,0)=0;

u(0,t)=u(6,t)=0

13.28.

utt =uxx

u(x,0)=0;

ut(x,0)= 3πsin3πx;

u(0,t)=u(5,t)=0

13.29.

utt =4uxx

u(x,0)=29sin2πx;

ut(x,0)=0;

u(0,t)=u(7,t)=0

13.30.

utt =9uxx

u(x,0)=0;

ut(x,0)= 6πsin2πx;

u(0,t)=u(6,t)=0

13.31.

utt =16uxx

u(x,0)=31sinπx;

ut(x,0)=0;

u(0,t)=u(8,t)=0