vpa(a2-a1) vpa(a3-a1)
vpa(a4-a1) %为精确值 vpa(a5-a1) vpa(a6-a1) vpa(a7-a1) vpa(a8-a1)
ans =
8.772689107613377606024459313047548287536202098197290121158158175e-72 ans =
8.772689107613377606024459313047548287536202098197290121158158175e-72 ans = 0.0 ans =
-0.0000000000000008874822716959584619522637254014249128254875650208152937300697045 ans =
-0.000000000000001489122128176563341755713716272780778030227615022223735634526288 ans =
-0.000000000000001518855593927822635897082947744411794950714383466168364259064934 ans =
-0.00000000000000151859755909122793880734918235619076228065004813152159311456667
可以看到，除了a4为精确，其余均存在很小的误差。其中a2与a3的误差较小，小于eps精度，故可认为为精确的。
3 独立自由变量
a1=sym('sin(w*t)') ; a2=sym('a*exp(-X)' ); a3=sym('z*exp(j*th)'); symvar(a1,1) symvar(a2,1) symvar(a3,1)
ans = w
ans = a
ans = z
6 符号解
syms x k; f1=x.^k;
s1=symsum(f1,k,0,inf); s2=subs(f1,x,(-1/3)); s3=subs(f1,x,(1/pi)); s4=subs(f1,x,3); symsum(s2,k,0,inf)
double(symsum(s3,k,0,inf)) symsum(s4,k,0,inf)
ans = 3/4 ans =
1.4669 ans = Inf
7 限定性假设
reset(symengine); syms k;
syms x positive;
f1=(2/(2*k+1))*((x-1)/(x+1))^(2*k+1); f1_s=symsum(f1,k,0,inf);
simplify(f1_s,'steps',27,'IgnoreAnalyticConstraints',true)
ans = log(x)
8 符号计算
syms t;
yt=abs(sin(t)); dydt=diff(yt,t)
dydt0=limit(dydt,t,0,'left') dydtpi=subs(dydt,t,(pi/2))
dydt =
sign(sin(t))*cos(t) dydt0 = -1
dydtpi = 0
9 积分值
syms x;
fx=exp(-abs(x))*abs(sin(x)) fxint=int(fx,-5*pi,1.7*pi); vpa(fxint,64)
fx =
abs(sin(x))*exp(-x) ans =
3617514.635647088707100018393465500554242735057835123431773680704
10二重积分
syms x y; fxy=x^2+y^2;
int(int(fxy,y,1,x^2),x,1,2)
ans =
1006/105
11 绘出曲线
syms t x;
fx=int((sin(t)./t),t,0,x); ezplot(fx)
fx4=subs(fx,x,4.5)
fx4 =
sinint(9/2) sinint(x)21.510.50-0.5-1-1.5-2-6-4-20x246
12 积分表达式
syms x;
syms n positive;
yn=int((sin(x)).^n,x,0,pi/2) yn3=subs(yn,n,1/3); vpa(yn3,32)
yn =
beta(1/2, n/2 + 1/2)/2 ans =
1.2935547796148952674767575125656
13 序列卷积
syms a b n;
syms k positive; xk=a.^k; hk=b.^k;
kn=subs(xk,k,k-n)*subs(hk,k,n); yk=symsum(kn,n,0,k)
yk =
piecewise([a == b and b ~= 0, b^k*(k + 1)], [a ~= b or b == 0, (a*a^k - b*b^k)/(a - b)])
所以答案为a*a^k - b*b^k)/(a - b)
20求解solve
reset(symengine) syms x y;
s=solve('x^2+y^2-1','x*y-2','x','y') s.x s.y
s =
x: [4x1 sym] y: [4x1 sym] ans =
((15^(1/2)*i)/2 + 1/2)^(1/2)/2 - ((15^(1/2)*i)/2 + 1/2)^(3/2)/2 - ((15^(1/2)*i)/2 + 1/2)^(1/2)/2 + ((15^(1/2)*i)/2 + 1/2)^(3/2)/2 (1/2 - (15^(1/2)*i)/2)^(1/2)/2 - (1/2 - (15^(1/2)*i)/2)^(3/2)/2 - (1/2 - (15^(1/2)*i)/2)^(1/2)/2 + (1/2 - (15^(1/2)*i)/2)^(3/2)/2 ans =
((15^(1/2)*i)/2 + 1/2)^(1/2) -((15^(1/2)*i)/2 + 1/2)^(1/2) (1/2 - (15^(1/2)*i)/2)^(1/2) -(1/2 - (15^(1/2)*i)/2)^(1/2)
23 求通解
clear all;
yso=simplify(dsolve('Dy*y*0.1+0.3*x=0','x'))
yso =
(- 3*x^2 + 2*C3)^(1/2) -(- 3*x^2 + 2*C3)^(1/2) %此题存疑
hold on;clear all; reset(symengine); syms x;
y1=(- 3*x^2 + 2*1)^(1/2); y2=-(- 3*x^2 + 2*1)^(1/2);
h1=ezplot(y1,x,[-2 2 -2 2],1); h2=ezplot(y2,x,[-2 2 -2 2],1);
grid on;title('');warning off;axis([-2 2 -2 2]); set(h1,'color','r','linewidth',2); set(h2,'color','r','linewidth',2); xlabel('Y');ylabel('x');
21.510.50-0.5-1-1.5-2-2x-1.5-1-0.50Y0.511.52
%对于第二章存在问题的习题的探讨 2.23
clear all; syms x;
yso=simplify(dsolve('Dy*y*0.1+0.3*x=0','x'))
%此题存疑
hold on;clear all; reset(symengine); syms x;
y1=(- 3*x^2 + 2*1)^(1/2); y2=-(- 3*x^2 + 2*1)^(1/2); h1=ezplot(y1,x,[-2 2 -2 2],1); h2=ezplot(y2,x,[-2 2 -2 2],1);
grid on;title('');warning off;axis([-2 2 -2 2]); set(h1,'color','r','linewidth',2); set(h2,'color','r','linewidth',2); xlabel('Y');ylabel('x');
yso =
(- 3*x^2 + 2*C3)^(1/2) -(- 3*x^2 + 2*C3)^(1/2)
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