实验十一 级数与方程符号求解.docx
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实验十一级数与方程符号求解
实验十一级数与方程符号求解
一、实验目的
1、掌握级数求和的方法
2、掌握将函数展开为泰勒级数的方法
3、掌握微分方程符号求解的方法
4、掌握代数方程符号求解的方法
二、实验内容
1、级数符号求和
(1)计算
>>n=sym('n');
s=symsum(1/(2*n-1),1,10)
s=
31037876/14549535
(2)求级数
的和函数,并求
之和。
>>n=sym('n');
x=sym('x');
s=symsum(n*n*x^(n-1),n,1,inf)
s=
(-x-1)/(x-1)^3
>>s1=symsum(n*n/5^n,n,1,inf)
s1=
15/32
2、将
在
处按5次多项式展开为泰勒级数。
>>x=sym('x');
taylor(log(x),6,1)
ans=
x-1-1/2*(x-1)^2+1/3*(x-1)^3-1/4*(x-1)^4+1/5*(x-1)^5
3、求下列方程的符号解。
(1)>>x=solve('log(1+x)-5/(1+sin(x)-2)','x')
x=
-3.9222501966216401268562035062927-1.5308457578423990317770669453365*i
(2)>>x=solve('x*x+9*sqrt(x+1)-1','x')
x=
-1+(1/12*(972+12*6465^(1/2))^(1/3)+2/(972+12*6465^(1/2))^(1/3)+1/2*i*3^(1/2)*(-1/6*(972+12*6465^(1/2))^(1/3)+4/(972+12*6465^(1/2))^(1/3)))^2
-1+(1/12*(972+12*6465^(1/2))^(1/3)+2/(972+12*6465^(1/2))^(1/3)-1/2*i*3^(1/2)*(-1/6*(972+12*6465^(1/2))^(1/3)+4/(972+12*6465^(1/2))^(1/3)))^2
-1
(3)>>x=solve('3*x*exp(x)+5*sin(x)-78.5','x')
x=
1.1438132465904248768705131138349-3.5140576459458125696291745801210*i
(4)>>[xy]=solve('sqrt(x*x+y*y)-100','3*x+5*y-8','x,y')
x=
12/17-10/17*21246^(1/2)
12/17+10/17*21246^(1/2)
y=
20/17+6/17*21246^(1/2)
20/17-6/17*21246^(1/2)
4、求微分方程初值问题的符号解,并与数值解进行比较。
>>functiondy=vdp29(t,y)
dy=zeros(2,1);
dy
(1)=y
(2);
dy
(2)=-4*y
(2)-29*y
(1);
[ty]=ode45('vdp29',[010],[0,15])
t=
0
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0001
0.0001
0.0002
0.0002
0.0003
0.0004
0.0008
0.0013
0.0017
0.0021
0.0042
0.0063
0.0084
0.0105
0.0209
0.0314
0.0419
0.0523
0.0892
0.1260
0.1628
0.1996
0.2377
0.2757
0.3138
0.3518
0.3893
0.4269
0.4644
0.5019
0.5397
0.5775
0.6153
0.6531
0.6908
0.7284
0.7660
0.8037
0.8416
0.8795
0.9174
0.9553
0.9928
1.0303
1.0678
1.1052
1.1430
1.1807
1.2185
1.2563
1.2939
1.3315
1.3691
1.4067
1.4445
1.4824
1.5203
1.5581
1.5961
1.6341
1.6721
1.7101
1.7478
1.7855
1.8232
1.8609
1.8985
1.9360
1.9736
2.0112
2.0490
2.0868
2.1246
2.1625
2.2007
2.2389
2.2772
2.3154
2.3531
2.3907
2.4284
2.4661
2.5036
2.5411
2.5786
2.6161
2.6539
2.6917
2.7295
2.7672
2.8054
2.8435
2.8816
2.9198
2.9574
2.9950
3.0326
3.0703
3.1077
3.1452
3.1826
3.2201
3.2578
3.2955
3.3333
3.3710
3.4091
3.4471
3.4852
3.5232
3.5608
3.5984
3.6360
3.6735
3.7114
3.7492
3.7871
3.8249
3.8626
3.9003
3.9380
3.9757
4.0137
4.0517
4.0896
4.1276
4.1659
4.2043
4.2426
4.2809
4.3201
4.3594
4.3986
4.4378
4.4754
4.5131
4.5507
4.5883
4.6263
4.6642
4.7022
4.7401
4.7870
4.8339
4.8808
4.9276
4.9694
5.0111
5.0529
5.0946
5.1336
5.1725
5.2114
5.2503
5.2947
5.3390
5.3834
5.4277
5.4909
5.5542
5.6174
5.6806
5.7319
5.7832
5.8344
5.8857
5.9390
5.9923
6.0456
6.0989
6.1636
6.2283
6.2930
6.3577
6.4224
6.4871
6.5519
6.6166
6.6869
6.7572
6.8276
6.8979
6.9949
7.0919
7.1889
7.2859
7.3678
7.4498
7.5317
7.6136
7.7317
7.8499
7.9680
8.0861
8.1968
8.3075
8.4183
8.5290
8.6726
8.8163
8.9599
9.1036
9.3004
9.4971
9.6939
9.8907
9.9180
9.9453
9.9727
10.0000
y=
015.0000
0.000114.9998
0.000114.9996
0.000214.9994
0.000214.9992
0.000514.9982
0.000714.9972
0.001014.9962
0.001214.9952
0.002514.9902
0.003714.9851
0.005014.9801
0.006214.9751
0.012514.9499
0.018714.9247
0.025014.8995
0.031214.8742
0.062214.7473
0.093014.6197
0.123414.4912
0.153614.3620
0.300513.7054
0.440513.0330
0.573312.3476
0.698911.6520
1.08229.1573
1.37366.6709
1.57494.2848
1.69122.0697
1.72990.0192
1.6960-1.7426
1.6014-3.1866
1.4581-4.3028
1.2807-5.0868
1.0796-5.5694
0.8662-5.7745
0.6497-5.7327
0.4371-5.4768
0.2376-5.0471
0.0572-4.4854
-0.1000-3.8308
-0.2310-3.1229
-0.3350-2.3954
-0.4116-1.6795
-0.4618-0.9999
-0.4876-0.3723
-0.49090.1813
-0.47490.6495
-0.44291.0267
-0.39881.3087
-0.34581.5022
-0.28721.6122
-0.22601.6461
-0.16431.6126
-0.10491.5221
-0.04991.3864
-0.00071.2168
0.04151.0249
0.07620.8209
0.10320.6146
0.12250.4138
0.13450.2238
0.13970.0519
0.1387-0.0976
0.1326-0.2223
0.1222-0.3214
0.1085-0.3940
0.0926-0.4412
0.0753-0.4646
0.0577-0.4668
0.0403-0.4508
0.0239-0.4200
0.0088-0.3775
-0.0044-0.3267
-0.0157-0.2706
-0.0247-0.2121
-0.0316-0.1538
-0.0363-0.0974
-0.0390-0.0451
-0.03980.0015
-0.03900.0415
-0.03670.0746
-0.03340.1001
-0.02920.1181
-0.02440.1289
-0.01950.1331
-0.01450.1316
-0.00960.1254
-0.00510.1152
-0.00100.1021
0.00250.0870
0.00550.0708
0.00780.0541
0.00960.0376
0.01070.0219
0.01120.0077
0.0113-0.0049
0.0109-0.0156
0.0101-0.0242
0.0091-0.0306
0.0078-0.0350
0.0065-0.0373
0.0050-0.0379
0.0036-0.0369
0.0023-0.0347
0.0010-0.0315
-0.0001-0.0276
-0.0010-0.0232
-0.0018-0.0185
-0.0024-0.0137
-0.0029-0.0091
-0.0031-0.0048
-0.0032-0.0009
-0.00320.0025
-0.00300.0054
-0.00280.0076
-0.00250.0092
-0.00210.0102
-0.00170.0107
-0.00130.0107
-0.00090.0103
-0.00050.0096
-0.00020.0086
0.00010.0074
0.00040.0061
0.00060.0047
0.00070.0034
0.00080.0021
0.00090.0009
0.0009-0.0001
0.0009-0.0010
0.0008-0.0018
0.0008-0.0023
0.0007-0.0027
0.0006-0.0030
0.0004-0.0031
0.0003-0.0030
0.0002-0.0029
0.0001-0.0026
0.0000-0.0023
-0.0001-0.0019
-0.0001-0.0015
-0.0002-0.0011
-0.0002-0.0008
-0.0003-0.0004
-0.0003-0.0001
-0.00030.0002
-0.00020.0004
-0.00020.0006
-0.00020.0008
-0.00020.0008
-0.00010.0009
-0.00010.0008
-0.00000.0008
-0.00000.0007
0.00000.0006
0.00000.0005
0.00010.0003
0.00010.0002
0.00010.0001
0.00010.0000
0.0001-0.0001
0.0001-0.0001
0.0001-0.0002
0.0001-0.0002
0.0000-0.0002
0.0000-0.0002
0.0000-0.0002
-0.0000-0.0002
-0.0000-0.0001
-0.0000-0.0001
-0.0000-0.0000
-0.0000-0.0000
-0.00000.0000
-0.00000.0000
-0.00000.0001
-0.00000.0001
-0.00000.0001
-0.00000.0001
0.00000.0000
0.00000.0000
0.00000.0000
0.00000.0000
0.0000-0.0000
0.0000-0.0000
0.0000-0.0000
0.0000-0.0000
0.0000-0.0000
0.0000-0.0000
-0.0000-0.0000
-0.0000-0.0000
-0.00000.0000
-0.00000.0000
-0.00000.0000
-0.00000.0000
-0.00000.0000
0.00000.0000
0.00000.0000
0.0000-0.0000
0.0000-0.0000
0.0000-0.0000
-0.0000-0.0000
-0.00000.0000
-0.00000.0000
-0.00000.0000
0.00000.0000
0.00000.0000
0.0000-0.0000
-0.0000-0.0000
-0.00000.0000
-0.00000.0000
0.00000.0000
0.0000-0.0000
0.0000-0.0000
0.0000-0.0000
0.0000-0.0000
0.0000-0.0000
>>plot(t,y(:
1),'-')
5、求微分方程组的通解。
>>[x,y,z]=dsolve('Dx=2*x-3*y+3*z','Dy=4*x-5*y+3*z','Dz=4*x-4*y+2*z','t')
x=
C2*exp(-t)+C3*exp(2*t)
y=
C2*exp(-t)+C3*exp(2*t)+exp(-2*t)*C1
z=
C3*exp(2*t)+exp(-2*t)*C1