matlab知识题及答案解析

2. 用MATLAB 语句输入矩阵A 和B
3．假设已知矩阵A ，试给出相应的MATLAB 命令，将其全部偶数行提取出来，
赋给B 矩阵，用magic(8)A =命令生成A 矩阵，用上述命令检验一下结果是不是正确。

4．用数值方法可以求出∑=++++++==63
63622284212i i S ，试不采用循环的
形式求出和式的数值解。

由于数值方法是采用double 形式进行计算的，难以保证有效位数字，所以结果不一定精确。

试采用运算的方法求该和式的精确值。

5．选择合适的步距绘制出下面的图形。

（1）)/1sin(t ，其中)1,1(-∈t ； （2）)tan(sin )sin(tan t t -，其中),(ππ-∈t
6. 试绘制出二元函数2
2
2
2
)1(1)1(1),(y
x y
x y x f z +++
+-=
=的三维图和三
视图
7. 试求出如下极限。

（1）x
x
x
x 1)93(lim +∞
→； （2）1
1lim
0-+→→xy xy y x ； （3）2
2)()cos(1lim
2
2
220
0y x y x e
y x y x +→→++-
8. 已知参数方程⎩⎨⎧-==t
t t y t x sin cos cos ln ，试求出x y d d 和3
/2
2d d π=t x y
9. 假设⎰-=xy
t t e y x f 0
d ),(2
，试求2
22222y f
y x f x f y x ∂∂+
∂∂∂-∂∂ 10. 试求出下面的极限。

（1）⎥⎦⎤⎢⎣⎡-++-+-+-∞→1)2(1
161141121lim 2222n n ； （2）)131211(
lim 2222π
πππn n n n n n n ++++++++∞
→ 11. 试求出以下的曲线积分。

（1）⎰+l
s y x d )(22，l 为曲线)sin (cos t t t a x +=，)cos (sin t t t a y -=，
)20(π≤≤t 。

（2）⎰-+++l
y y y xe x e yx )dy 2(xy d )(33，其中l 为22222c y b x a =+正向上半
椭圆。

12. 试求出Vandermonde 矩阵⎥⎥⎥⎥⎥
⎥⎦
⎤⎢⎢⎢⎢⎢⎢⎣⎡=1e
e e e 1d d d d 1c c c c 1b b b b
1a a a a 2
34234234
234234A 的行列式，并以最简的形式显示结果。

13. 试对矩阵⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎢⎣⎡-------=22120.54.50.520.50.51.500.50.50.52A 进行Jordan 变换，并得出变换矩阵。

14. 试用数值方法和解析方法求取下面的Sylvester 方程，并验证得出的结果。

⎥⎥
⎥
⎥⎥
⎥⎦
⎤
⎢⎢⎢⎢⎢⎢⎣⎡-------=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-----+⎥⎥⎥⎥⎥
⎥⎦
⎤
⎢⎢⎢
⎢⎢⎢⎣⎡-------36644616521411291229212343040
01101013376364224150463
X X
15. 假设已知矩阵A 如下，试求出At e ，At sin ，)sin(2t e A e At At 。

⎥⎥⎥⎥
⎦
⎤⎢⎢⎢
⎢⎣⎡---------=3110 1.52.511.50.50.540.5 1.50.50
4.5A 第二部分数学问题求解与数据处理(4 学时)
主要问题：掌握代数方程与最优化问题、微分方程问题、数据处理问题的MATLAB
求解方法。

1. 对下列的函数)(t f 进行Laplace 变换。

（1）t
t
t f a αsin )(=
；（2）t t t f b αsin )(5=；（3）t t t f c αcos )(8= 2. 对下面的)(s F 式进行Laplace 反变换。

（1）))((1)(2
2
2
b s a s s s F a +-=；（2）b s a s s F b ---=)(；
（3）b
s a
s s F c --=ln
)(。

3. 试求出下面函数的Fourier 变换，对得出的结果再进行Fourier 反变换，观
察是否能得出原来函数。

（1）ππ20),23()(2≤≤-=x x x x f ；（2）ππ20,)2()(22≤≤-=t t t t f 。

4. 请将下述时域序列函数)(kT f 进行Z 变换，并对结果进行反变换检验。

（1）)cos()(kaT kT f a =；（2）akT b e kT kT f -=2)()(；（3）)1(1
)(akT c e akT a
kT f -+-=
5. 用数值求解函数求解下述一元和二元方程的根，并对得出的结果进行检验。

（1）)25sin(2
/)1()(2+++-=x x e
x f π；（2）xy
y x e
xy y x y x f ---++=22)(),(2
2
6. 试求出使得⎰-1
02d )(x cx e x 取得极小值的c 值。

7. 试求解下面的非线性规划问题。

min )12424(2212
2
211++++x x x x x e x x ⎪⎪⎩⎪⎪⎨⎧≤≤--≥≥++-≤+10
,10105.10
.s.t 21212
12121x x x x x x x x x x
8. 求解下面的整数线性规划问题。

max )23374855273381592(7654321x x x x x x x ++++++
x ⎩⎨⎧≤++++++≥119567
3044515285891767235635340
.s.t 7654321x x x x x x x x
9. 试求出微分方程x e x x y x
x y x x y 52)()11()()1
2()(-=-+-- 的解析解通解，并求出满足边界条件1)(,)1(==ππy y 的解析解。

10. 试求出下面微分方程的通解。

（1）1)()(2)(2+=++t t x t t x t t x ；（2）2
)(2)(x xe x xy x y
-=+ 11. 考虑著名的ssler o
R 化学反应方程组⎪⎩⎪
⎨⎧-+=+=--=z
c x b z
ay x y z y x
)( ，选定2.0==b a ，7.5=c ，且)0()0()0(321x x x ==，绘制仿真结果的三维相轨迹，并得出其在x-y 平面上的投影。

在实际求解中建议将c b a ,,作为附加参数，同样的方程若设
2.0=a ，5.0=b ，10=c 时，绘制出状态变量的二维图和三维图。

12. 试选择状态变量，将下面的非线性微分方程组转换成一阶显式微分方程组，
并用 MATLAB 对其求解，绘制出解的相平面或相空间曲线。

⎪⎪⎩
⎪⎪⎨
⎧==-===----=+++---=-6)1(,7)1(,2)1(4)1(,2)1(26)()3()
3(32y y y x
x t e x y y t y y x y x x x
13.考虑简单的线性微分方程)3/4sin(246553)3()4(π++=++++--t e e y y y y y t t
，且方程的初值为1)0(=y ，2/1)0()0(==y
y ，2.0)0()3(=y ，试用Simulink 搭建起系统的仿真模型，并绘制出仿真结果曲线。

14. 用t e t t y t sin )(52-=生成一组较稀疏的数据，并用一维数据插值的方法对给
出的数据进行曲线拟合，并将结果与理论曲线相比较。

第一部分
第二题
（1）
>> A=[1,2,3,4;4,3,2,1;2,3,4,1;3,2,4,1]
A =
1 2 3 4
4 3 2 1
2 3 4 1
3 2
4 1
（2）
>>
B=[1+4j,2+3j,3+2j,4+1j;4+1j,3+2j,2+3j,1+4j;2+3j,3+2j,4+1j,1+4j;3+2j,2+3j,4+1j,1+ 4j]
B =
1.0000 + 4.0000i
2.0000 +
3.0000i 3.0000 + 2.0000i
4.0000 + 1.0000i
4.0000 + 1.0000i 3.0000 + 2.0000i 2.0000 + 3.0000i 1.0000 + 4.0000i
2.0000 +
3.0000i 3.0000 + 2.0000i
4.0000 + 1.0000i 1.0000 + 4.0000i
3.0000 + 2.0000i 2.0000 + 3.0000i
4.0000 + 1.0000i 1.0000 + 4.0000i 第三题
>> A=magic(8);
>> B=A(2:2:end,:)
B =
9 55 54 12 13 51 50 16
40 26 27 37 36 30 31 33
41 23 22 44 45 19 18 48
8 58 59 5 4 62 63 1 第四题
>> i=0:63;s=sum(2.^i)
s =
1.8447e+019
第五题
（1）
>> t=[-1:0.001:1];
>> y=sin(1./t);
Warning: Divide by zero.
>> plot(t,y)
（2）
t=[-pi:0.05:-1.8,-1.799:0.001:-1.2,-1.2:0.05:1.2,1.201:0.001:1.8,1.81:0.05:pi]; >> y=sin(tan(t))-tan(sin(t));
>> plot(t,y)
第六题
>> xx=[-2:0.1:-1.2,-1.1:0.02:-0.9,-0.8:0.1:0.8,0.9:0.02:1.1,1.2:0.1:2]; >> yy=[-1:0.1:-0.2,-0.1:0.02:0.1,0.2:0.1:1];[x,y]=meshgrid(xx,yy); >> z=1./(sqrt((1-x).^2+y.^2))+1./(sqrt((1+x).^2+y.^2)); Warning: Divide by zero.
Warning: Divide by zero.
>> subplot(224),surf(x,y,z)
>> subplot(221),surf(x,y,z),view(0,90)
>> subplot(222),surf(x,y,z),view(90,0)
>> subplot(223),surf(x,y,z),view(0,0)
第七题
（1）>> syms x;f=(3^x+9^x)^(1/x);l=limit(f,x,inf)
l =
9
（2）>> syms x y;f=x*y/(sqrt(x*y+1)-1);limit(limit(f,x,0),y,0) ans =
2
（3）>> syms x y;f=(1-cos(x^2+y^2))*exp(x^2+y^2)/(x^2+y^2);limit(limit(f,x,0),y,0) ans =
第八题
>> syms t;x=log(cos(t));y=cos(t)-t*sin(t);diff(y,t)/diff(x,t)
ans =
-(-2*sin(t)-t*cos(t))/sin(t)*cos(t)
>> f=diff(y,t,2)/diff(x,t,2);subs(f,t,sym(pi)/3)
ans =
3/8-1/24*pi*3^(1/2)
第九题
>> syms x y t
>> s=int(exp(-t^2),t,0,x*y);
>> x/y*diff(f,x,2)-2*diff(diff(f,x),y)+diff(f,y,2)
ans =
2*x^2*y^2*exp(-x^2*y^2)-2*exp(-x^2*y^2)-2*x^3*y*exp(-x^2*y^2)
第十题
（1）
>> syms k n;symsum(1/((2*k)^2-1),k,1,inf)
ans =
1/2
>> limit(symsum(1/((2*k)^2-1),k,1,n),n,inf)
ans =
1/2
（2）
>> limit(n*symsum(1/(n^2+k*pi),k,1,n),n,inf)
ans =
1
第十一题
（1）
>> syms a t;x=a*(cos(t)+t*sin(t));y=a*(sin(t)-t*cos(t)); >> f=x^2+y^2;I=int(f*sqrt(diff(x,t)^2+diff(y,t)^2),t,0,2*pi)
I =
2*a^2*pi^2*(a^2)^(1/2)+4*a^2*pi^4*(a^2)^(1/2)
（2）
>> syms x y a b c t;x=c*cos(t)/a;y=c*sin(t)/b;
>> P=y*x^3+exp(y);Q=x*y^3+x*exp(y)-2*y;
>> ds=[diff(x,t);diff(y,t)];I=int([P Q]*ds,t,0,pi)
I =
-2/15*c*(-2*c^4+15*b^4)/b^4/a
第十二题
>> syms a b c d e;A=vander([a b c d e])
A =
[ a^4, a^3, a^2, a, 1]
[ b^4, b^3, b^2, b, 1]
[ c^4, c^3, c^2, c, 1]
[ d^4, d^3, d^2, d, 1]
[ e^4, e^3, e^2, e, 1]
>> det(A),simple(ans)
ans =
a^4*b^3*c^2*d-a^4*b^3*c^2*e-a^4*b^3*d^2*c+a^4*b^3*d^2*e+a^4*b^3*e^2 *c-a^4*b^3*e^2*d-a^4*c^3*b^2*d+a^4*c^3*b^2*e+a^4*c^3*d^2*b-a^4*c^3*d ^2*e-a^4*c^3*e^2*b+a^4*c^3*e^2*d+a^4*d^3*b^2*c-a^4*d^3*b^2*e-a^4*d^3 *c^2*b+a^4*d^3*c^2*e+a^4*d^3*e^2*b-a^4*d^3*e^2*c-a^4*e^3*b^2*c+a^4*e ^3*b^2*d+a^4*e^3*c^2*b-a^4*e^3*c^2*d-a^4*e^3*d^2*b+a^4*e^3*d^2*c-b^ 4*a^3*c^2*d+b^4*a^3*c^2*e+b^4*a^3*d^2*c-b^4*a^3*d^2*e-b^4*a^3*e^2*c+ b^4*a^3*e^2*d+b^4*c^3*a^2*d-b^4*c^3*a^2*e-b^4*c^3*d^2*a+b^4*c^3*d^2 *e+b^4*c^3*e^2*a-b^4*c^3*e^2*d-b^4*d^3*a^2*c+b^4*d^3*a^2*e+b^4*d^3*c ^2*a-b^4*d^3*c^2*e-b^4*d^3*e^2*a+b^4*d^3*e^2*c+b^4*e^3*a^2*c-b^4*e^ 3*a^2*d-b^4*e^3*c^2*a+b^4*e^3*c^2*d+b^4*e^3*d^2*a-b^4*e^3*d^2*c+c^4
*a^3*b^2*d-c^4*a^3*b^2*e-c^4*a^3*d^2*b+c^4*a^3*d^2*e+c^4*a^3*e^2*b-c ^4*a^3*e^2*d-c^4*b^3*a^2*d+c^4*b^3*a^2*e+c^4*b^3*d^2*a-c^4*b^3*d^2* e-c^4*b^3*e^2*a+c^4*b^3*e^2*d+c^4*d^3*a^2*b-c^4*d^3*a^2*e-c^4*d^3*b ^2*a+c^4*d^3*b^2*e+c^4*d^3*e^2*a-c^4*d^3*e^2*b-c^4*e^3*a^2*b+c^4*e^ 3*a^2*d+c^4*e^3*b^2*a-c^4*e^3*b^2*d-c^4*e^3*d^2*a+c^4*e^3*d^2*b-d^4* a^3*b^2*c+d^4*a^3*b^2*e+d^4*a^3*c^2*b-d^4*a^3*c^2*e-d^4*a^3*e^2*b+d ^4*a^3*e^2*c+d^4*b^3*a^2*c-d^4*b^3*a^2*e-d^4*b^3*c^2*a+d^4*b^3*c^2* e+d^4*b^3*e^2*a-d^4*b^3*e^2*c-d^4*c^3*a^2*b+d^4*c^3*a^2*e+d^4*c^3*b ^2*a-d^4*c^3*b^2*e-d^4*c^3*e^2*a+d^4*c^3*e^2*b+d^4*e^3*a^2*b-d^4*e^ 3*a^2*c-d^4*e^3*b^2*a+d^4*e^3*b^2*c+d^4*e^3*c^2*a-d^4*e^3*c^2*b+e^4 *a^3*b^2*c-e^4*a^3*b^2*d-e^4*a^3*c^2*b+e^4*a^3*c^2*d+e^4*a^3*d^2*b-e ^4*a^3*d^2*c-e^4*b^3*a^2*c+e^4*b^3*a^2*d+e^4*b^3*c^2*a-e^4*b^3*c^2* d-e^4*b^3*d^2*a+e^4*b^3*d^2*c+e^4*c^3*a^2*b-e^4*c^3*a^2*d-e^4*c^3*b ^2*a+e^4*c^3*b^2*d+e^4*c^3*d^2*a-e^4*c^3*d^2*b-e^4*d^3*a^2*b+e^4*d^ 3*a^2*c+e^4*d^3*b^2*a-e^4*d^3*b^2*c-e^4*d^3*c^2*a+e^4*d^3*c^2*b
simplify:
a^4*b^3*c^2*d-a^4*b^3*c^2*e-a^4*b^3*d^2*c+a^4*b^3*d^2*e+a^4*b^3*e^2 *c-a^4*b^3*e^2*d-a^4*c^3*b^2*d+a^4*c^3*b^2*e+a^4*c^3*d^2*b-a^4*c^3*d
^2*e-a^4*c^3*e^2*b+a^4*c^3*e^2*d+a^4*d^3*b^2*c-a^4*d^3*b^2*e-a^4*d^3 *c^2*b+a^4*d^3*c^2*e+a^4*d^3*e^2*b-a^4*d^3*e^2*c-a^4*e^3*b^2*c+a^4*e ^3*b^2*d+a^4*e^3*c^2*b-a^4*e^3*c^2*d-a^4*e^3*d^2*b+a^4*e^3*d^2*c-b^ 4*a^3*c^2*d+b^4*a^3*c^2*e+b^4*a^3*d^2*c-b^4*a^3*d^2*e-b^4*a^3*e^2*c+ b^4*a^3*e^2*d+b^4*c^3*a^2*d-b^4*c^3*a^2*e-b^4*c^3*d^2*a+b^4*c^3*d^2 *e+b^4*c^3*e^2*a-b^4*c^3*e^2*d-b^4*d^3*a^2*c+b^4*d^3*a^2*e+b^4*d^3*c ^2*a-b^4*d^3*c^2*e-b^4*d^3*e^2*a+b^4*d^3*e^2*c+b^4*e^3*a^2*c-b^4*e^ 3*a^2*d-b^4*e^3*c^2*a+b^4*e^3*c^2*d+b^4*e^3*d^2*a-b^4*e^3*d^2*c+c^4 *a^3*b^2*d-c^4*a^3*b^2*e-c^4*a^3*d^2*b+c^4*a^3*d^2*e+c^4*a^3*e^2*b-c ^4*a^3*e^2*d-c^4*b^3*a^2*d+c^4*b^3*a^2*e+c^4*b^3*d^2*a-c^4*b^3*d^2* e-c^4*b^3*e^2*a+c^4*b^3*e^2*d+c^4*d^3*a^2*b-c^4*d^3*a^2*e-c^4*d^3*b ^2*a+c^4*d^3*b^2*e+c^4*d^3*e^2*a-c^4*d^3*e^2*b-c^4*e^3*a^2*b+c^4*e^ 3*a^2*d+c^4*e^3*b^2*a-c^4*e^3*b^2*d-c^4*e^3*d^2*a+c^4*e^3*d^2*b-d^4* a^3*b^2*c+d^4*a^3*b^2*e+d^4*a^3*c^2*b-d^4*a^3*c^2*e-d^4*a^3*e^2*b+d ^4*a^3*e^2*c+d^4*b^3*a^2*c-d^4*b^3*a^2*e-d^4*b^3*c^2*a+d^4*b^3*c^2* e+d^4*b^3*e^2*a-d^4*b^3*e^2*c-d^4*c^3*a^2*b+d^4*c^3*a^2*e+d^4*c^3*b ^2*a-d^4*c^3*b^2*e-d^4*c^3*e^2*a+d^4*c^3*e^2*b+d^4*e^3*a^2*b-d^4*e^ 3*a^2*c-d^4*e^3*b^2*a+d^4*e^3*b^2*c+d^4*e^3*c^2*a-d^4*e^3*c^2*b+e^4 *a^3*b^2*c-e^4*a^3*b^2*d-e^4*a^3*c^2*b+e^4*a^3*c^2*d+e^4*a^3*d^2*b-e ^4*a^3*d^2*c-e^4*b^3*a^2*c+e^4*b^3*a^2*d+e^4*b^3*c^2*a-e^4*b^3*c^2* d-e^4*b^3*d^2*a+e^4*b^3*d^2*c+e^4*c^3*a^2*b-e^4*c^3*a^2*d-e^4*c^3*b ^2*a+e^4*c^3*b^2*d+e^4*c^3*d^2*a-e^4*c^3*d^2*b-e^4*d^3*a^2*b+e^4*d^
3*a^2*c+e^4*d^3*b^2*a-e^4*d^3*b^2*c-e^4*d^3*c^2*a+e^4*d^3*c^2*b radsimp:
a^4*b^3*c^2*d-a^4*b^3*c^2*e-a^4*b^3*d^2*c+a^4*b^3*d^2*e+a^4*b^3*e^2 *c-a^4*b^3*e^2*d-a^4*c^3*b^2*d+a^4*c^3*b^2*e+a^4*c^3*d^2*b-a^4*c^3*d ^2*e-a^4*c^3*e^2*b+a^4*c^3*e^2*d+a^4*d^3*b^2*c-a^4*d^3*b^2*e-a^4*d^3 *c^2*b+a^4*d^3*c^2*e+a^4*d^3*e^2*b-a^4*d^3*e^2*c-a^4*e^3*b^2*c+a^4*e ^3*b^2*d+a^4*e^3*c^2*b-a^4*e^3*c^2*d-a^4*e^3*d^2*b+a^4*e^3*d^2*c-b^ 4*a^3*c^2*d+b^4*a^3*c^2*e+b^4*a^3*d^2*c-b^4*a^3*d^2*e-b^4*a^3*e^2*c+ b^4*a^3*e^2*d+b^4*c^3*a^2*d-b^4*c^3*a^2*e-b^4*c^3*d^2*a+b^4*c^3*d^2 *e+b^4*c^3*e^2*a-b^4*c^3*e^2*d-b^4*d^3*a^2*c+b^4*d^3*a^2*e+b^4*d^3*c ^2*a-b^4*d^3*c^2*e-b^4*d^3*e^2*a+b^4*d^3*e^2*c+b^4*e^3*a^2*c-b^4*e^ 3*a^2*d-b^4*e^3*c^2*a+b^4*e^3*c^2*d+b^4*e^3*d^2*a-b^4*e^3*d^2*c+c^4 *a^3*b^2*d-c^4*a^3*b^2*e-c^4*a^3*d^2*b+c^4*a^3*d^2*e+c^4*a^3*e^2*b-c ^4*a^3*e^2*d-c^4*b^3*a^2*d+c^4*b^3*a^2*e+c^4*b^3*d^2*a-c^4*b^3*d^2* e-c^4*b^3*e^2*a+c^4*b^3*e^2*d+c^4*d^3*a^2*b-c^4*d^3*a^2*e-c^4*d^3*b ^2*a+c^4*d^3*b^2*e+c^4*d^3*e^2*a-c^4*d^3*e^2*b-c^4*e^3*a^2*b+c^4*e^ 3*a^2*d+c^4*e^3*b^2*a-c^4*e^3*b^2*d-c^4*e^3*d^2*a+c^4*e^3*d^2*b-d^4* a^3*b^2*c+d^4*a^3*b^2*e+d^4*a^3*c^2*b-d^4*a^3*c^2*e-d^4*a^3*e^2*b+d ^4*a^3*e^2*c+d^4*b^3*a^2*c-d^4*b^3*a^2*e-d^4*b^3*c^2*a+d^4*b^3*c^2*
e+d^4*b^3*e^2*a-d^4*b^3*e^2*c-d^4*c^3*a^2*b+d^4*c^3*a^2*e+d^4*c^3*b ^2*a-d^4*c^3*b^2*e-d^4*c^3*e^2*a+d^4*c^3*e^2*b+d^4*e^3*a^2*b-d^4*e^ 3*a^2*c-d^4*e^3*b^2*a+d^4*e^3*b^2*c+d^4*e^3*c^2*a-d^4*e^3*c^2*b+e^4 *a^3*b^2*c-e^4*a^3*b^2*d-e^4*a^3*c^2*b+e^4*a^3*c^2*d+e^4*a^3*d^2*b-e ^4*a^3*d^2*c-e^4*b^3*a^2*c+e^4*b^3*a^2*d+e^4*b^3*c^2*a-e^4*b^3*c^2* d-e^4*b^3*d^2*a+e^4*b^3*d^2*c+e^4*c^3*a^2*b-e^4*c^3*a^2*d-e^4*c^3*b ^2*a+e^4*c^3*b^2*d+e^4*c^3*d^2*a-e^4*c^3*d^2*b-e^4*d^3*a^2*b+e^4*d^ 3*a^2*c+e^4*d^3*b^2*a-e^4*d^3*b^2*c-e^4*d^3*c^2*a+e^4*d^3*c^2*b
combine(trig):
a^4*b^3*c^2*d-a^4*b^3*c^2*e-a^4*b^3*d^2*c+a^4*b^3*d^2*e+a^4*b^3*e^2 *c-a^4*b^3*e^2*d-a^4*c^3*b^2*d+a^4*c^3*b^2*e+a^4*c^3*d^2*b-a^4*c^3*d ^2*e-a^4*c^3*e^2*b+a^4*c^3*e^2*d+a^4*d^3*b^2*c-a^4*d^3*b^2*e-a^4*d^3 *c^2*b+a^4*d^3*c^2*e+a^4*d^3*e^2*b-a^4*d^3*e^2*c-a^4*e^3*b^2*c+a^4*e ^3*b^2*d+a^4*e^3*c^2*b-a^4*e^3*c^2*d-a^4*e^3*d^2*b+a^4*e^3*d^2*c-b^ 4*a^3*c^2*d+b^4*a^3*c^2*e+b^4*a^3*d^2*c-b^4*a^3*d^2*e-b^4*a^3*e^2*c+ b^4*a^3*e^2*d+b^4*c^3*a^2*d-b^4*c^3*a^2*e-b^4*c^3*d^2*a+b^4*c^3*d^2 *e+b^4*c^3*e^2*a-b^4*c^3*e^2*d-b^4*d^3*a^2*c+b^4*d^3*a^2*e+b^4*d^3*c ^2*a-b^4*d^3*c^2*e-b^4*d^3*e^2*a+b^4*d^3*e^2*c+b^4*e^3*a^2*c-b^4*e^ 3*a^2*d-b^4*e^3*c^2*a+b^4*e^3*c^2*d+b^4*e^3*d^2*a-b^4*e^3*d^2*c+c^4
*a^3*b^2*d-c^4*a^3*b^2*e-c^4*a^3*d^2*b+c^4*a^3*d^2*e+c^4*a^3*e^2*b-c ^4*a^3*e^2*d-c^4*b^3*a^2*d+c^4*b^3*a^2*e+c^4*b^3*d^2*a-c^4*b^3*d^2* e-c^4*b^3*e^2*a+c^4*b^3*e^2*d+c^4*d^3*a^2*b-c^4*d^3*a^2*e-c^4*d^3*b ^2*a+c^4*d^3*b^2*e+c^4*d^3*e^2*a-c^4*d^3*e^2*b-c^4*e^3*a^2*b+c^4*e^ 3*a^2*d+c^4*e^3*b^2*a-c^4*e^3*b^2*d-c^4*e^3*d^2*a+c^4*e^3*d^2*b-d^4* a^3*b^2*c+d^4*a^3*b^2*e+d^4*a^3*c^2*b-d^4*a^3*c^2*e-d^4*a^3*e^2*b+d ^4*a^3*e^2*c+d^4*b^3*a^2*c-d^4*b^3*a^2*e-d^4*b^3*c^2*a+d^4*b^3*c^2* e+d^4*b^3*e^2*a-d^4*b^3*e^2*c-d^4*c^3*a^2*b+d^4*c^3*a^2*e+d^4*c^3*b ^2*a-d^4*c^3*b^2*e-d^4*c^3*e^2*a+d^4*c^3*e^2*b+d^4*e^3*a^2*b-d^4*e^ 3*a^2*c-d^4*e^3*b^2*a+d^4*e^3*b^2*c+d^4*e^3*c^2*a-d^4*e^3*c^2*b+e^4 *a^3*b^2*c-e^4*a^3*b^2*d-e^4*a^3*c^2*b+e^4*a^3*c^2*d+e^4*a^3*d^2*b-e ^4*a^3*d^2*c-e^4*b^3*a^2*c+e^4*b^3*a^2*d+e^4*b^3*c^2*a-e^4*b^3*c^2* d-e^4*b^3*d^2*a+e^4*b^3*d^2*c+e^4*c^3*a^2*b-e^4*c^3*a^2*d-e^4*c^3*b ^2*a+e^4*c^3*b^2*d+e^4*c^3*d^2*a-e^4*c^3*d^2*b-e^4*d^3*a^2*b+e^4*d^ 3*a^2*c+e^4*d^3*b^2*a-e^4*d^3*b^2*c-e^4*d^3*c^2*a+e^4*d^3*c^2*b
factor:
(c-d)*(b-d)*(b-c)*(a-d)*(a-c)*(a-b)*(-d+e)*(e-c)*(e-b)*(e-a)
expand:
a^4*b^3*c^2*d-a^4*b^3*c^2*e-a^4*b^3*d^2*c+a^4*b^3*d^2*e+a^4*b^3*e^2 *c-a^4*b^3*e^2*d-a^4*c^3*b^2*d+a^4*c^3*b^2*e+a^4*c^3*d^2*b-a^4*c^3*d ^2*e-a^4*c^3*e^2*b+a^4*c^3*e^2*d+a^4*d^3*b^2*c-a^4*d^3*b^2*e-a^4*d^3 *c^2*b+a^4*d^3*c^2*e+a^4*d^3*e^2*b-a^4*d^3*e^2*c-a^4*e^3*b^2*c+a^4*e ^3*b^2*d+a^4*e^3*c^2*b-a^4*e^3*c^2*d-a^4*e^3*d^2*b+a^4*e^3*d^2*c-b^ 4*a^3*c^2*d+b^4*a^3*c^2*e+b^4*a^3*d^2*c-b^4*a^3*d^2*e-b^4*a^3*e^2*c+ b^4*a^3*e^2*d+b^4*c^3*a^2*d-b^4*c^3*a^2*e-b^4*c^3*d^2*a+b^4*c^3*d^2 *e+b^4*c^3*e^2*a-b^4*c^3*e^2*d-b^4*d^3*a^2*c+b^4*d^3*a^2*e+b^4*d^3*c ^2*a-b^4*d^3*c^2*e-b^4*d^3*e^2*a+b^4*d^3*e^2*c+b^4*e^3*a^2*c-b^4*e^ 3*a^2*d-b^4*e^3*c^2*a+b^4*e^3*c^2*d+b^4*e^3*d^2*a-b^4*e^3*d^2*c+c^4 *a^3*b^2*d-c^4*a^3*b^2*e-c^4*a^3*d^2*b+c^4*a^3*d^2*e+c^4*a^3*e^2*b-c ^4*a^3*e^2*d-c^4*b^3*a^2*d+c^4*b^3*a^2*e+c^4*b^3*d^2*a-c^4*b^3*d^2* e-c^4*b^3*e^2*a+c^4*b^3*e^2*d+c^4*d^3*a^2*b-c^4*d^3*a^2*e-c^4*d^3*b ^2*a+c^4*d^3*b^2*e+c^4*d^3*e^2*a-c^4*d^3*e^2*b-c^4*e^3*a^2*b+c^4*e^ 3*a^2*d+c^4*e^3*b^2*a-c^4*e^3*b^2*d-c^4*e^3*d^2*a+c^4*e^3*d^2*b-d^4* a^3*b^2*c+d^4*a^3*b^2*e+d^4*a^3*c^2*b-d^4*a^3*c^2*e-d^4*a^3*e^2*b+d ^4*a^3*e^2*c+d^4*b^3*a^2*c-d^4*b^3*a^2*e-d^4*b^3*c^2*a+d^4*b^3*c^2* e+d^4*b^3*e^2*a-d^4*b^3*e^2*c-d^4*c^3*a^2*b+d^4*c^3*a^2*e+d^4*c^3*b ^2*a-d^4*c^3*b^2*e-d^4*c^3*e^2*a+d^4*c^3*e^2*b+d^4*e^3*a^2*b-d^4*e^ 3*a^2*c-d^4*e^3*b^2*a+d^4*e^3*b^2*c+d^4*e^3*c^2*a-d^4*e^3*c^2*b+e^4
*a^3*b^2*c-e^4*a^3*b^2*d-e^4*a^3*c^2*b+e^4*a^3*c^2*d+e^4*a^3*d^2*b-e ^4*a^3*d^2*c-e^4*b^3*a^2*c+e^4*b^3*a^2*d+e^4*b^3*c^2*a-e^4*b^3*c^2* d-e^4*b^3*d^2*a+e^4*b^3*d^2*c+e^4*c^3*a^2*b-e^4*c^3*a^2*d-e^4*c^3*b ^2*a+e^4*c^3*b^2*d+e^4*c^3*d^2*a-e^4*c^3*d^2*b-e^4*d^3*a^2*b+e^4*d^ 3*a^2*c+e^4*d^3*b^2*a-e^4*d^3*b^2*c-e^4*d^3*c^2*a+e^4*d^3*c^2*b
combine:
a^4*b^3*c^2*d-a^4*b^3*c^2*e-a^4*b^3*d^2*c+a^4*b^3*d^2*e+a^4*b^3*e^2 *c-a^4*b^3*e^2*d-a^4*c^3*b^2*d+a^4*c^3*b^2*e+a^4*c^3*d^2*b-a^4*c^3*d ^2*e-a^4*c^3*e^2*b+a^4*c^3*e^2*d+a^4*d^3*b^2*c-a^4*d^3*b^2*e-a^4*d^3 *c^2*b+a^4*d^3*c^2*e+a^4*d^3*e^2*b-a^4*d^3*e^2*c-a^4*e^3*b^2*c+a^4*e ^3*b^2*d+a^4*e^3*c^2*b-a^4*e^3*c^2*d-a^4*e^3*d^2*b+a^4*e^3*d^2*c-b^ 4*a^3*c^2*d+b^4*a^3*c^2*e+b^4*a^3*d^2*c-b^4*a^3*d^2*e-b^4*a^3*e^2*c+ b^4*a^3*e^2*d+b^4*c^3*a^2*d-b^4*c^3*a^2*e-b^4*c^3*d^2*a+b^4*c^3*d^2 *e+b^4*c^3*e^2*a-b^4*c^3*e^2*d-b^4*d^3*a^2*c+b^4*d^3*a^2*e+b^4*d^3*c ^2*a-b^4*d^3*c^2*e-b^4*d^3*e^2*a+b^4*d^3*e^2*c+b^4*e^3*a^2*c-b^4*e^ 3*a^2*d-b^4*e^3*c^2*a+b^4*e^3*c^2*d+b^4*e^3*d^2*a-b^4*e^3*d^2*c+c^4 *a^3*b^2*d-c^4*a^3*b^2*e-c^4*a^3*d^2*b+c^4*a^3*d^2*e+c^4*a^3*e^2*b-c ^4*a^3*e^2*d-c^4*b^3*a^2*d+c^4*b^3*a^2*e+c^4*b^3*d^2*a-c^4*b^3*d^2* e-c^4*b^3*e^2*a+c^4*b^3*e^2*d+c^4*d^3*a^2*b-c^4*d^3*a^2*e-c^4*d^3*b
^2*a+c^4*d^3*b^2*e+c^4*d^3*e^2*a-c^4*d^3*e^2*b-c^4*e^3*a^2*b+c^4*e^ 3*a^2*d+c^4*e^3*b^2*a-c^4*e^3*b^2*d-c^4*e^3*d^2*a+c^4*e^3*d^2*b-d^4* a^3*b^2*c+d^4*a^3*b^2*e+d^4*a^3*c^2*b-d^4*a^3*c^2*e-d^4*a^3*e^2*b+d ^4*a^3*e^2*c+d^4*b^3*a^2*c-d^4*b^3*a^2*e-d^4*b^3*c^2*a+d^4*b^3*c^2* e+d^4*b^3*e^2*a-d^4*b^3*e^2*c-d^4*c^3*a^2*b+d^4*c^3*a^2*e+d^4*c^3*b ^2*a-d^4*c^3*b^2*e-d^4*c^3*e^2*a+d^4*c^3*e^2*b+d^4*e^3*a^2*b-d^4*e^ 3*a^2*c-d^4*e^3*b^2*a+d^4*e^3*b^2*c+d^4*e^3*c^2*a-d^4*e^3*c^2*b+e^4 *a^3*b^2*c-e^4*a^3*b^2*d-e^4*a^3*c^2*b+e^4*a^3*c^2*d+e^4*a^3*d^2*b-e ^4*a^3*d^2*c-e^4*b^3*a^2*c+e^4*b^3*a^2*d+e^4*b^3*c^2*a-e^4*b^3*c^2* d-e^4*b^3*d^2*a+e^4*b^3*d^2*c+e^4*c^3*a^2*b-e^4*c^3*a^2*d-e^4*c^3*b ^2*a+e^4*c^3*b^2*d+e^4*c^3*d^2*a-e^4*c^3*d^2*b-e^4*d^3*a^2*b+e^4*d^ 3*a^2*c+e^4*d^3*b^2*a-e^4*d^3*b^2*c-e^4*d^3*c^2*a+e^4*d^3*c^2*b
convert(exp):
a^4*b^3*c^2*d-a^4*b^3*c^2*e-a^4*b^3*d^2*c+a^4*b^3*d^2*e+a^4*b^3*e^2 *c-a^4*b^3*e^2*d-a^4*c^3*b^2*d+a^4*c^3*b^2*e+a^4*c^3*d^2*b-a^4*c^3*d ^2*e-a^4*c^3*e^2*b+a^4*c^3*e^2*d+a^4*d^3*b^2*c-a^4*d^3*b^2*e-a^4*d^3 *c^2*b+a^4*d^3*c^2*e+a^4*d^3*e^2*b-a^4*d^3*e^2*c-a^4*e^3*b^2*c+a^4*e ^3*b^2*d+a^4*e^3*c^2*b-a^4*e^3*c^2*d-a^4*e^3*d^2*b+a^4*e^3*d^2*c-b^ 4*a^3*c^2*d+b^4*a^3*c^2*e+b^4*a^3*d^2*c-b^4*a^3*d^2*e-b^4*a^3*e^2*c+
b^4*a^3*e^2*d+b^4*c^3*a^2*d-b^4*c^3*a^2*e-b^4*c^3*d^2*a+b^4*c^3*d^2 *e+b^4*c^3*e^2*a-b^4*c^3*e^2*d-b^4*d^3*a^2*c+b^4*d^3*a^2*e+b^4*d^3*c ^2*a-b^4*d^3*c^2*e-b^4*d^3*e^2*a+b^4*d^3*e^2*c+b^4*e^3*a^2*c-b^4*e^ 3*a^2*d-b^4*e^3*c^2*a+b^4*e^3*c^2*d+b^4*e^3*d^2*a-b^4*e^3*d^2*c+c^4 *a^3*b^2*d-c^4*a^3*b^2*e-c^4*a^3*d^2*b+c^4*a^3*d^2*e+c^4*a^3*e^2*b-c ^4*a^3*e^2*d-c^4*b^3*a^2*d+c^4*b^3*a^2*e+c^4*b^3*d^2*a-c^4*b^3*d^2* e-c^4*b^3*e^2*a+c^4*b^3*e^2*d+c^4*d^3*a^2*b-c^4*d^3*a^2*e-c^4*d^3*b ^2*a+c^4*d^3*b^2*e+c^4*d^3*e^2*a-c^4*d^3*e^2*b-c^4*e^3*a^2*b+c^4*e^ 3*a^2*d+c^4*e^3*b^2*a-c^4*e^3*b^2*d-c^4*e^3*d^2*a+c^4*e^3*d^2*b-d^4* a^3*b^2*c+d^4*a^3*b^2*e+d^4*a^3*c^2*b-d^4*a^3*c^2*e-d^4*a^3*e^2*b+d ^4*a^3*e^2*c+d^4*b^3*a^2*c-d^4*b^3*a^2*e-d^4*b^3*c^2*a+d^4*b^3*c^2* e+d^4*b^3*e^2*a-d^4*b^3*e^2*c-d^4*c^3*a^2*b+d^4*c^3*a^2*e+d^4*c^3*b ^2*a-d^4*c^3*b^2*e-d^4*c^3*e^2*a+d^4*c^3*e^2*b+d^4*e^3*a^2*b-d^4*e^ 3*a^2*c-d^4*e^3*b^2*a+d^4*e^3*b^2*c+d^4*e^3*c^2*a-d^4*e^3*c^2*b+e^4 *a^3*b^2*c-e^4*a^3*b^2*d-e^4*a^3*c^2*b+e^4*a^3*c^2*d+e^4*a^3*d^2*b-e ^4*a^3*d^2*c-e^4*b^3*a^2*c+e^4*b^3*a^2*d+e^4*b^3*c^2*a-e^4*b^3*c^2* d-e^4*b^3*d^2*a+e^4*b^3*d^2*c+e^4*c^3*a^2*b-e^4*c^3*a^2*d-e^4*c^3*b ^2*a+e^4*c^3*b^2*d+e^4*c^3*d^2*a-e^4*c^3*d^2*b-e^4*d^3*a^2*b+e^4*d^ 3*a^2*c+e^4*d^3*b^2*a-e^4*d^3*b^2*c-e^4*d^3*c^2*a+e^4*d^3*c^2*b
convert(sincos):
a^4*b^3*c^2*d-a^4*b^3*c^2*e-a^4*b^3*d^2*c+a^4*b^3*d^2*e+a^4*b^3*e^2 *c-a^4*b^3*e^2*d-a^4*c^3*b^2*d+a^4*c^3*b^2*e+a^4*c^3*d^2*b-a^4*c^3*d ^2*e-a^4*c^3*e^2*b+a^4*c^3*e^2*d+a^4*d^3*b^2*c-a^4*d^3*b^2*e-a^4*d^3 *c^2*b+a^4*d^3*c^2*e+a^4*d^3*e^2*b-a^4*d^3*e^2*c-a^4*e^3*b^2*c+a^4*e ^3*b^2*d+a^4*e^3*c^2*b-a^4*e^3*c^2*d-a^4*e^3*d^2*b+a^4*e^3*d^2*c-b^ 4*a^3*c^2*d+b^4*a^3*c^2*e+b^4*a^3*d^2*c-b^4*a^3*d^2*e-b^4*a^3*e^2*c+ b^4*a^3*e^2*d+b^4*c^3*a^2*d-b^4*c^3*a^2*e-b^4*c^3*d^2*a+b^4*c^3*d^2 *e+b^4*c^3*e^2*a-b^4*c^3*e^2*d-b^4*d^3*a^2*c+b^4*d^3*a^2*e+b^4*d^3*c ^2*a-b^4*d^3*c^2*e-b^4*d^3*e^2*a+b^4*d^3*e^2*c+b^4*e^3*a^2*c-b^4*e^ 3*a^2*d-b^4*e^3*c^2*a+b^4*e^3*c^2*d+b^4*e^3*d^2*a-b^4*e^3*d^2*c+c^4 *a^3*b^2*d-c^4*a^3*b^2*e-c^4*a^3*d^2*b+c^4*a^3*d^2*e+c^4*a^3*e^2*b-c ^4*a^3*e^2*d-c^4*b^3*a^2*d+c^4*b^3*a^2*e+c^4*b^3*d^2*a-c^4*b^3*d^2* e-c^4*b^3*e^2*a+c^4*b^3*e^2*d+c^4*d^3*a^2*b-c^4*d^3*a^2*e-c^4*d^3*b ^2*a+c^4*d^3*b^2*e+c^4*d^3*e^2*a-c^4*d^3*e^2*b-c^4*e^3*a^2*b+c^4*e^ 3*a^2*d+c^4*e^3*b^2*a-c^4*e^3*b^2*d-c^4*e^3*d^2*a+c^4*e^3*d^2*b-d^4* a^3*b^2*c+d^4*a^3*b^2*e+d^4*a^3*c^2*b-d^4*a^3*c^2*e-d^4*a^3*e^2*b+d ^4*a^3*e^2*c+d^4*b^3*a^2*c-d^4*b^3*a^2*e-d^4*b^3*c^2*a+d^4*b^3*c^2* e+d^4*b^3*e^2*a-d^4*b^3*e^2*c-d^4*c^3*a^2*b+d^4*c^3*a^2*e+d^4*c^3*b ^2*a-d^4*c^3*b^2*e-d^4*c^3*e^2*a+d^4*c^3*e^2*b+d^4*e^3*a^2*b-d^4*e^ 3*a^2*c-d^4*e^3*b^2*a+d^4*e^3*b^2*c+d^4*e^3*c^2*a-d^4*e^3*c^2*b+e^4 *a^3*b^2*c-e^4*a^3*b^2*d-e^4*a^3*c^2*b+e^4*a^3*c^2*d+e^4*a^3*d^2*b-e
^4*a^3*d^2*c-e^4*b^3*a^2*c+e^4*b^3*a^2*d+e^4*b^3*c^2*a-e^4*b^3*c^2* d-e^4*b^3*d^2*a+e^4*b^3*d^2*c+e^4*c^3*a^2*b-e^4*c^3*a^2*d-e^4*c^3*b ^2*a+e^4*c^3*b^2*d+e^4*c^3*d^2*a-e^4*c^3*d^2*b-e^4*d^3*a^2*b+e^4*d^ 3*a^2*c+e^4*d^3*b^2*a-e^4*d^3*b^2*c-e^4*d^3*c^2*a+e^4*d^3*c^2*b
convert(tan):
a^4*b^3*c^2*d-a^4*b^3*c^2*e-a^4*b^3*d^2*c+a^4*b^3*d^2*e+a^4*b^3*e^2 *c-a^4*b^3*e^2*d-a^4*c^3*b^2*d+a^4*c^3*b^2*e+a^4*c^3*d^2*b-a^4*c^3*d ^2*e-a^4*c^3*e^2*b+a^4*c^3*e^2*d+a^4*d^3*b^2*c-a^4*d^3*b^2*e-a^4*d^3 *c^2*b+a^4*d^3*c^2*e+a^4*d^3*e^2*b-a^4*d^3*e^2*c-a^4*e^3*b^2*c+a^4*e ^3*b^2*d+a^4*e^3*c^2*b-a^4*e^3*c^2*d-a^4*e^3*d^2*b+a^4*e^3*d^2*c-b^ 4*a^3*c^2*d+b^4*a^3*c^2*e+b^4*a^3*d^2*c-b^4*a^3*d^2*e-b^4*a^3*e^2*c+ b^4*a^3*e^2*d+b^4*c^3*a^2*d-b^4*c^3*a^2*e-b^4*c^3*d^2*a+b^4*c^3*d^2 *e+b^4*c^3*e^2*a-b^4*c^3*e^2*d-b^4*d^3*a^2*c+b^4*d^3*a^2*e+b^4*d^3*c ^2*a-b^4*d^3*c^2*e-b^4*d^3*e^2*a+b^4*d^3*e^2*c+b^4*e^3*a^2*c-b^4*e^ 3*a^2*d-b^4*e^3*c^2*a+b^4*e^3*c^2*d+b^4*e^3*d^2*a-b^4*e^3*d^2*c+c^4 *a^3*b^2*d-c^4*a^3*b^2*e-c^4*a^3*d^2*b+c^4*a^3*d^2*e+c^4*a^3*e^2*b-c ^4*a^3*e^2*d-c^4*b^3*a^2*d+c^4*b^3*a^2*e+c^4*b^3*d^2*a-c^4*b^3*d^2* e-c^4*b^3*e^2*a+c^4*b^3*e^2*d+c^4*d^3*a^2*b-c^4*d^3*a^2*e-c^4*d^3*b ^2*a+c^4*d^3*b^2*e+c^4*d^3*e^2*a-c^4*d^3*e^2*b-c^4*e^3*a^2*b+c^4*e^
3*a^2*d+c^4*e^3*b^2*a-c^4*e^3*b^2*d-c^4*e^3*d^2*a+c^4*e^3*d^2*b-d^4* a^3*b^2*c+d^4*a^3*b^2*e+d^4*a^3*c^2*b-d^4*a^3*c^2*e-d^4*a^3*e^2*b+d ^4*a^3*e^2*c+d^4*b^3*a^2*c-d^4*b^3*a^2*e-d^4*b^3*c^2*a+d^4*b^3*c^2* e+d^4*b^3*e^2*a-d^4*b^3*e^2*c-d^4*c^3*a^2*b+d^4*c^3*a^2*e+d^4*c^3*b ^2*a-d^4*c^3*b^2*e-d^4*c^3*e^2*a+d^4*c^3*e^2*b+d^4*e^3*a^2*b-d^4*e^ 3*a^2*c-d^4*e^3*b^2*a+d^4*e^3*b^2*c+d^4*e^3*c^2*a-d^4*e^3*c^2*b+e^4 *a^3*b^2*c-e^4*a^3*b^2*d-e^4*a^3*c^2*b+e^4*a^3*c^2*d+e^4*a^3*d^2*b-e ^4*a^3*d^2*c-e^4*b^3*a^2*c+e^4*b^3*a^2*d+e^4*b^3*c^2*a-e^4*b^3*c^2* d-e^4*b^3*d^2*a+e^4*b^3*d^2*c+e^4*c^3*a^2*b-e^4*c^3*a^2*d-e^4*c^3*b ^2*a+e^4*c^3*b^2*d+e^4*c^3*d^2*a-e^4*c^3*d^2*b-e^4*d^3*a^2*b+e^4*d^ 3*a^2*c+e^4*d^3*b^2*a-e^4*d^3*b^2*c-e^4*d^3*c^2*a+e^4*d^3*c^2*b
collect(e):
a^4*b^3*c^2*d-a^4*b^3*d^2*c-a^4*c^3*b^2*d+a^4*c^3*d^2*b+a^4*d^3*b^2 *c-a^4*d^3*c^2*b-b^4*a^3*c^2*d+b^4*a^3*d^2*c+b^4*c^3*a^2*d-b^4*c^3*d ^2*a-b^4*d^3*a^2*c+b^4*d^3*c^2*a+c^4*a^3*b^2*d-c^4*a^3*d^2*b-c^4*b^ 3*a^2*d+c^4*b^3*d^2*a+c^4*d^3*a^2*b-c^4*d^3*b^2*a-d^4*a^3*b^2*c+d^4 *a^3*c^2*b+d^4*b^3*a^2*c-d^4*b^3*c^2*a-d^4*c^3*a^2*b+d^4*c^3*b^2*a+ (a^3*b^2*c-a^3*b^2*d-a^3*c^2*b+a^3*c^2*d+a^3*d^2*b-a^3*d^2*c-b^3*a^2 *c+b^3*a^2*d+b^3*c^2*a-b^3*c^2*d-b^3*d^2*a+b^3*d^2*c+c^3*a^2*b-c^3*
a^2*d-c^3*b^2*a+c^3*b^2*d+c^3*d^2*a-c^3*d^2*b-d^3*a^2*b+d^3*a^2*c+d ^3*b^2*a-d^3*b^2*c-d^3*c^2*a+d^3*c^2*b)*e^4+(-a^4*b^2*c-c^4*d*b^2-c^4 *d^2*a+c^4*d^2*b+a^4*b^2*d+a^4*c^2*b-a^4*c^2*d-c^4*a^2*b+c^4*a^2*d+ c^4*b^2*a+b^4*a^2*c-b^4*a^2*d-b^4*c^2*a+c^2*d*b^4+b^4*d^2*a-d^2*c*b ^4-a^4*d^2*b+a^4*d^2*c+d^4*a^2*b-d^4*a^2*c-d^4*b^2*a+c*d^4*b^2+d^4 *c^2*a-c^2*d^4*b)*e^3+(b^4*a^3*d-c^4*b^3*a-a^4*c^3*b-c*d^4*b^3+c^4*d* b^3-a^4*b^3*d-b^4*a^3*c+a^4*c^3*d+a^4*b^3*c-a^4*d^3*c-c^3*d*b^4+d^4 *b^3*a+a^4*d^3*b+b^4*c^3*a-c^4*a^3*d-d^4*a^3*b+c^4*d^3*a+c^4*a^3*b-c^4*d^3*b-b^4*d^3*a+c*d^3*b^4+d^4*a^3*c-d^4*c^3*a+c^3*d^4*b)*e^2+(a ^4*b^3*d^2+c^4*a^3*d^2-b^4*c^3*a^2-a^4*b^3*c^2+a^4*d^3*c^2+d^4*c^3 *a^2-c^4*d^2*b^3-a^4*c^3*d^2+c^4*d^3*b^2+c^4*b^3*a^2-c^4*d^3*a^2-a ^4*d^3*b^2-b^4*a^3*d^2+b^4*d^3*a^2+b^4*a^3*c^2-c^3*d^4*b^2-d^4*a^ 3*c^2+c^2*d^4*b^3+a^4*c^3*b^2+c^3*d^2*b^4-c^2*d^3*b^4-c^4*a^3*b^2 +d^4*a^3*b^2-d^4*b^3*a^2)*e
mwcos2sin:
a^4*b^3*c^2*d-a^4*b^3*c^2*e-a^4*b^3*d^2*c+a^4*b^3*d^2*e+a^4*b^3*e^2 *c-a^4*b^3*e^2*d-a^4*c^3*b^2*d+a^4*c^3*b^2*e+a^4*c^3*d^2*b-a^4*c^3*d ^2*e-a^4*c^3*e^2*b+a^4*c^3*e^2*d+a^4*d^3*b^2*c-a^4*d^3*b^2*e-a^4*d^3 *c^2*b+a^4*d^3*c^2*e+a^4*d^3*e^2*b-a^4*d^3*e^2*c-a^4*e^3*b^2*c+a^4*e
^3*b^2*d+a^4*e^3*c^2*b-a^4*e^3*c^2*d-a^4*e^3*d^2*b+a^4*e^3*d^2*c-b^ 4*a^3*c^2*d+b^4*a^3*c^2*e+b^4*a^3*d^2*c-b^4*a^3*d^2*e-b^4*a^3*e^2*c+ b^4*a^3*e^2*d+b^4*c^3*a^2*d-b^4*c^3*a^2*e-b^4*c^3*d^2*a+b^4*c^3*d^2 *e+b^4*c^3*e^2*a-b^4*c^3*e^2*d-b^4*d^3*a^2*c+b^4*d^3*a^2*e+b^4*d^3*c ^2*a-b^4*d^3*c^2*e-b^4*d^3*e^2*a+b^4*d^3*e^2*c+b^4*e^3*a^2*c-b^4*e^ 3*a^2*d-b^4*e^3*c^2*a+b^4*e^3*c^2*d+b^4*e^3*d^2*a-b^4*e^3*d^2*c+c^4 *a^3*b^2*d-c^4*a^3*b^2*e-c^4*a^3*d^2*b+c^4*a^3*d^2*e+c^4*a^3*e^2*b-c ^4*a^3*e^2*d-c^4*b^3*a^2*d+c^4*b^3*a^2*e+c^4*b^3*d^2*a-c^4*b^3*d^2* e-c^4*b^3*e^2*a+c^4*b^3*e^2*d+c^4*d^3*a^2*b-c^4*d^3*a^2*e-c^4*d^3*b ^2*a+c^4*d^3*b^2*e+c^4*d^3*e^2*a-c^4*d^3*e^2*b-c^4*e^3*a^2*b+c^4*e^ 3*a^2*d+c^4*e^3*b^2*a-c^4*e^3*b^2*d-c^4*e^3*d^2*a+c^4*e^3*d^2*b-d^4* a^3*b^2*c+d^4*a^3*b^2*e+d^4*a^3*c^2*b-d^4*a^3*c^2*e-d^4*a^3*e^2*b+d ^4*a^3*e^2*c+d^4*b^3*a^2*c-d^4*b^3*a^2*e-d^4*b^3*c^2*a+d^4*b^3*c^2* e+d^4*b^3*e^2*a-d^4*b^3*e^2*c-d^4*c^3*a^2*b+d^4*c^3*a^2*e+d^4*c^3*b ^2*a-d^4*c^3*b^2*e-d^4*c^3*e^2*a+d^4*c^3*e^2*b+d^4*e^3*a^2*b-d^4*e^ 3*a^2*c-d^4*e^3*b^2*a+d^4*e^3*b^2*c+d^4*e^3*c^2*a-d^4*e^3*c^2*b+e^4 *a^3*b^2*c-e^4*a^3*b^2*d-e^4*a^3*c^2*b+e^4*a^3*c^2*d+e^4*a^3*d^2*b-e ^4*a^3*d^2*c-e^4*b^3*a^2*c+e^4*b^3*a^2*d+e^4*b^3*c^2*a-e^4*b^3*c^2* d-e^4*b^3*d^2*a+e^4*b^3*d^2*c+e^4*c^3*a^2*b-e^4*c^3*a^2*d-e^4*c^3*b ^2*a+e^4*c^3*b^2*d+e^4*c^3*d^2*a-e^4*c^3*d^2*b-e^4*d^3*a^2*b+e^4*d^ 3*a^2*c+e^4*d^3*b^2*a-e^4*d^3*b^2*c-e^4*d^3*c^2*a+e^4*d^3*c^2*b
ans =
(c-d)*(b-d)*(b-c)*(a-d)*(a-c)*(a-b)*(-d+e)*(e-c)*(e-b)*(e-a)
>>
第十三题
>> A=[-2,0.5,-0.5,0.5;0,-1.5,0.5,-0.5;2,0.5,-4.5,0.5;2,1,-2,-2];[V J]=jordan(sym(A)) V =
[ 0, 1/2, 1/2, -1/4]
[ 0, 0, 1/2, 1]
[ 1/4, 1/2, 1/2, -1/4]
[ 1/4, 1/2, 1, -1/4]
J =
[ -4, 0, 0, 0]
[ 0, -2, 1, 0]
[ 0, 0, -2, 1]
[ 0, 0, 0, -2]
第十四题
数值方法
>> A=[3,-6,-4,0,5;1,4,2,-2,4;-6,3,-6,7,3;-13,10,0,-11,0;0,4,0,3,4]; >> B=[3,-2,1;-2,9,2;-2,-1,9];
>> C=[-2,1,-1;4,1,2;5,-6,1;6,-4,-4;-6,6,-3];
>> X=lyap(A,B,C)
X =
-2.3192 -0.4678 0.1505
-3.6284 0.1579 0.0629
5.4246 -1.0516 -0.5090
-0.5718 2.5848 -0.3649
3.0417 -0.6265 0.1580
>> norm(A*X+X*B+C)
ans =
3.8830e-014
解析方法
>> edit
function X=lyap(A,B,C)
if nargin==2,C=B;B=A';end
[nr,nc]=size(C);A0=kron(A,eye(nc))+kron(eye(nr),B');
try
C1=C';x0=-inv(A0)*C1(:);X=reshape(x0,nc,nr)';
catch,error('singular matrix found.'),end
>> A=[3,-6,-4,0,5;1,4,2,-2,4;-6,3,-6,7,3;-13,10,0,-11,0;0,4,0,3,4];
>> B=[3,-2,1;-2,-9,2;-2,-1,9];
>> C=[-2,1,-1;4,1,2;5,-6,1;6,-4,-4;-6,6,-3];X=lyap(sym(A),B,C)
X =
[ -434641749950/107136516451, -4664546747350/321409549353, 503105815912/321409549353]
[ 3809507498/107136516451, 8059112319373/321409549353, -880921527508/321409549353]
[ 1016580400173/107136516451, 8334897743767/321409549353, -1419901706449/321409549353]
[ 288938859984/107136516451, 6956912657222/321409549353, -927293592476/321409549353]
[ 827401644798/107136516451, 10256166034813/321409549353, -1209595497577/321409549353]
>> A*X+X*B+C
ans =
[ 0, 0, 0]
[ 0, 0, 0]
[ 0, 0, 0]
[ 0, 0, 0]
[ 0, 0, 0]
第十五题
（1）
>> A=[-4.5,0,0.5,-1.5;-0.5,-4,0.5,-0.5;1.5,1,-2.5,1.5;0,-1,-1,-3];
>> A=sym(A);syms t;
>> expm(A*t)
ans =
[ 1/2*exp(-3*t)-1/2*t*exp(-3*t)+1/2*exp(-5*t)+1/2*t^2*exp(-3*t), 1/2*exp(-5*t)-1/2*exp(-3*t)+t*exp(-3*t),
1/2*t*exp(-3*t)+1/2*t^2*exp(-3*t),
1/2*exp(-5*t)-1/2*exp(-3*t)-1/2*t*exp(-3*t)+1/2*t^2*exp(-3*t)]
[ 1/2*t*exp(-3*t)+1/2*exp(-5*t)-1/2*exp(-3*t), 1/2*exp(-3*t)+1/2*exp(-5*t),
1/2*t*exp(-3*t), 1/2*t*exp(-3*t)+1/2*exp(-5*t)-1/2*exp(-3*t)] [ 1/2*t*exp(-3*t)-1/2*exp(-5*t)+1/2*exp(-3*t), -1/2*exp(-5*t)+1/2*exp(-3*t),
exp(-3*t)+1/2*t*exp(-3*t),
1/2*t*exp(-3*t)-1/2*exp(-5*t)+1/2*exp(-3*t)]
[ -1/2*t^2*exp(-3*t), -t*exp(-3*t), -1/2*t^2*exp(-3*t)-t*exp(-3*t), exp(-3*t)-1/2*t^2*exp(-3*t)]
(2)
>> A=[-4.5,0,0.5,-1.5;-0.5,-4,0.5,-0.5;1.5,1,-2.5,1.5;0,-1,-1,-3];
>> A=sym(A);syms t;
>> sin(A*t)
ans =
[ -sin(9/2*t), 0, sin(1/2*t), -sin(3/2*t)]
[ -sin(1/2*t), -sin(4*t), sin(1/2*t), -sin(1/2*t)] [ sin(3/2*t), sin(t), -sin(5/2*t), sin(3/2*t)] [ 0, -sin(t), -sin(t), -sin(3*t)] (3)
第二部分
第一题
（1）
>> syms a t;f=sin(a*t)/t;laplace(f)
ans =
atan(a/s)
（2）
>> syms t a;f=t^5*sin(a*t);laplace(f)
ans =
60*i*(-1/(s-i*a)^6+1/(s+i*a)^6)
（3）
>> syms t a;f=t^8*cos(a*t);laplace(f)
ans =。