题目

二阶参数微分方程画图
二阶参数微分方程组:
d^2(x)/dt^2=nb{-Sin(wt)dz/dt+Cos(wt)dy/dt}
d^2(y)/dt^2=n{eSin(wt+Pi/2)-bCos(wt)dx/dt}
d^2(z)/dt^2=n{eCos(wt+Pi/2)+bSin(wt)dx/dt}
初始条件:
t=0,x=y=z=0,dx/dt=0,dy/dt=0,dz/dt=0
式中已知常数:
n=3034,b=0.4,e=57,w=25
求t在区间(0,0.1)的图形.
昨天自己用mathematica试过,软件解不出来.用mathematica或者matlab画图,把运行成功的代码发给我.

分类:


满意答案

观察方程组,发现可用降阶法,求出dx、dy、dz,再积分,求出x、y、z.
clc;clear;
n=3034,b=0.4,e=57,w=25
[dx,dy,dz]=dsolve('Dx=n*b*(-sin(w*t)*z+cos(w*t)*y),Dy=n*(e*sin(w*t+pi/2)-b*cos(w*t)*x),Dz=n*(e*cos(w*t+pi/2)+b*sin(w*t)*x)','x(0)=0,y(0)=0,z(0)=0')
dx=subs(dx),dy=subs(dy),dz=subs(dz)%代入参数
x=int(dx),y=int(dy),z=int(dz)%积分
t=0:0.005:0.1;
subplot(3,1,1),plot(t,subs(x),'o-'),ylabel('x')
subplot(3,1,2),plot(t,subs(y),'*r-'),ylabel('y')
subplot(3,1,3),plot(t,subs(z),'^g-'),ylabel('z'),xlabel('t')
figure
t=0:0.004:0.1;%奇怪,为什么间隔不同,图像也不同啊?
subplot(3,1,1),plot(t,subs(x),'o-'),ylabel('x')
subplot(3,1,2),plot(t,subs(y),'*r-'),ylabel('y')
subplot(3,1,3),plot(t,subs(z),'^g-'),ylabel('z'),xlabel('t')
结果:
n = 3034
b = 0.4000
e = 57
w = 25
dx =
n^2*b*e/(n^2*b^2+w^2)-n^2*b*e/(n^2*b^2+w^2)*cos((n^2*b^2+w^2)^(1/2)*t)
dy =
1/2*(2*n^2*sin(w*t)*b*e-2*n^4*sin(w*t)*b^3*e/(n^2*b^2+w^2)+n^2*b*e/(n^2*b^2+w^2)^(1/2)*w*sin(w*t+(n^2*b^2+w^2)^(1/2)*t)-n^2*b*e/(n^2*b^2+w^2)^(1/2)*w*sin(w*t-(n^2*b^2+w^2)^(1/2)*t)-w^2*n^2*b*e/(n^2*b^2+w^2)*sin(w*t+(n^2*b^2+w^2)^(1/2)*t)-w^2*n^2*b*e/(n^2*b^2+w^2)*sin(w*t-(n^2*b^2+w^2)^(1/2)*t))/b/n/w
dz =
1/2*(-n^2*b*e/(n^2*b^2+w^2)^(1/2)*w*cos(w*t-(n^2*b^2+w^2)^(1/2)*t)+n^2*b*e/(n^2*b^2+w^2)^(1/2)*w*cos(w*t+(n^2*b^2+w^2)^(1/2)*t)+2*n^2*b*e*cos(w*t)-w^2*n^2*b*e/(n^2*b^2+w^2)*cos(w*t-(n^2*b^2+w^2)^(1/2)*t)-w^2*n^2*b*e/(n^2*b^2+w^2)*cos(w*t+(n^2*b^2+w^2)^(1/2)*t)-2*n^4*b^3*e/(n^2*b^2+w^2)*cos(w*t))/b/n/w
dx =
5246938920/36836249-5246938920/36836249*cos(1/25*36836249^(1/2)*25^(1/2)*t)
dy =
108086250/36836249*sin(25*t)+86469/36836249*36836249^(1/2)*25^(1/2)*sin(25*t+1/25*36836249^(1/2)*25^(1/2)*t)+86469/36836249*36836249^(1/2)*25^(1/2)*sin(-25*t+1/25*36836249^(1/2)*25^(1/2)*t)-54043125/36836249*sin(25*t+1/25*36836249^(1/2)*25^(1/2)*t)+54043125/36836249*sin(-25*t+1/25*36836249^(1/2)*25^(1/2)*t)
dz =
108086250/36836249*cos(25*t)-54043125/36836249*cos(-25*t+1/25*36836249^(1/2)*25^(1/2)*t)-54043125/36836249*cos(25*t+1/25*36836249^(1/2)*25^(1/2)*t)-86469/36836249*36836249^(1/2)*25^(1/2)*cos(-25*t+1/25*36836249^(1/2)*25^(1/2)*t)+86469/36836249*36836249^(1/2)*25^(1/2)*cos(25*t+1/25*36836249^(1/2)*25^(1/2)*t)
x =
5246938920/36836249*t-5246938920/1356909240390001*sin(1/25*36836249^(1/2)*25^(1/2)*t)*36836249^(1/2)*25^(1/2)
y =
-54043125/36836249/(-25+1/25*36836249^(1/2)*25^(1/2))*cos((-25+1/25*36836249^(1/2)*25^(1/2))*t)-86469/36836249*36836249^(1/2)*25^(1/2)/(25+1/25*36836249^(1/2)*25^(1/2))*cos((25+1/25*36836249^(1/2)*25^(1/2))*t)-86469/36836249*36836249^(1/2)*25^(1/2)/(-25+1/25*36836249^(1/2)*25^(1/2))*cos((-25+1/25*36836249^(1/2)*25^(1/2))*t)+54043125/36836249/(25+1/25*36836249^(1/2)*25^(1/2))*cos((25+1/25*36836249^(1/2)*25^(1/2))*t)-4323450/36836249*cos(25*t)
z =
4323450/36836249*sin(25*t)-54043125/36836249*sin((-25+1/25*36836249^(1/2)*25^(1/2))*t)/(-25+1/25*36836249^(1/2)*25^(1/2))-54043125/36836249*sin((25+1/25*36836249^(1/2)*25^(1/2))*t)/(25+1/25*36836249^(1/2)*25^(1/2))-86469/36836249*36836249^(1/2)*25^(1/2)*sin((-25+1/25*36836249^(1/2)*25^(1/2))*t)/(-25+1/25*36836249^(1/2)*25^(1/2))+86469/36836249*36836249^(1/2)*25^(1/2)*sin((25+1/25*36836249^(1/2)*25^(1/2))*t)/(25+1/25*36836249^(1/2)*25^(1/2))

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=sin
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B
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B+C
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,即sin
B
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A+C
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2
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A
2
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C
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A
2
cos
C
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A
2
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C
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B+C
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sin
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2
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B
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2
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2
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2
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C
2

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A
2
tan
C
2
=
1
2

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B
2
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B
2

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A
2
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C
2
cos
B
2

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A
2
[sin
B+C
2
-sin
B−C
2
]
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1
2
sinA-cos
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2
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