MATLAB作业5

MATLAB 作业5

1、 试求出下面线性微分方程的通解。

543225432()()()()()136415217680()[sin(2)cos(3)]3

t

d y t d y t d y t d y t dy t y t

e t t dt dt dt dt dt π-+++++=++假设上述微分方程满足已知条件，(0)1,(1)3,()2,(0)1,(1)2y y y y y π=====试求出满足该条件的微分方程的解析解。 解： >> syms t y ;

y=dsolve(['D5y+13*D4y+64*D3y+152*D2y+176*Dy+80*y=','exp(-2*t)*(sin(2*t+pi/3)+cos (3*t))'],'y(0)=1','y(1)=3','y(pi)=2','Dy(0)=1','Dy(1)=2'); vpa(y,20) ans =

.20576131687242798354e-2*exp(-2.*t)*cos(3.*t)+.15538705805619602373e-1*exp(-2.*t)*sin (2.*t)+.76830587084294035587e-2*exp(-2.*t)*cos(2.*t)+98.159206062620455336*exp(-2.*t)*t+59.405044899367325899*exp(-2.*t)*t^3-106.24422608844727795*exp(-2.*t)*t^2-30.741892776456442810*exp(-2.*t)+.20576131687242798354e-2*exp(-2.*t)*sin(3.*t)+31.732152104579289128*exp(-5.*t)

2、 试求解下面微分方程的通解以及满足(0)1,()2,(0)0x x y π===条件下的解析解。

66()5()4()3()sin(4)

2()()4()6()cos(4)t t

x t x t x t y t e t y t y t x t x t e t --⎧+++=⎨+++=⎩

[x,y]=dsolve('D2x+5*Dx+4*x+3*y=exp(-6*t)*sin(4*t)','2*Dy+y+4*Dx+6*x=exp(-6*t)*cos(4*t)','x(0)=1','x(pi)=2','y(0)=0')； >> vpa(x,10) ans =

0.0858********exp(t) - 0.057658489325149275828152894973755/(exp(7.549834435*t)^(1/4)*exp(t)^(13/4)) + (0.9469805542*exp(7.549834435*t)^(1/4))/exp(t)^(13/4) + (0.024*********cos(4.0*t))/exp(6.0*t) - (0.016682998530139342028763560499272*sin(4.0*t))/exp(6.0*t)

>> vpa(y,10) ans = - 0.28620556196983670815825462341309*exp(t) + 0.09045056185/(exp(7.549834435*t)^(1/4)*exp(t)^(13/4)) + (0.3018304533*exp(7.549834435*t)^(1/4))/exp(t)^(13/4) - (0.10607545320921207832043364760466*cos(4.0*t))/exp(6.0*t) + (0.0683488486*sin(4.0*t))/exp(6.0*t)

3、 试求出微分方程2511()(2)()(1)()x

y x y x y x x e

x x

---+-=的解析解通解，并求出满足

边界条件(1),()1y y ππ==的解析解。

>> syms x y;

y=dsolve('D2y-(2-1/x)*Dy+(1-1/x)*y=x^2*exp(-5*x)') y =

C3*exp(t) + C2*exp((t*(x - 1))/x) + x^3/(exp(5*x)*(x - 1))

>> y=dsolve('D2y-(2-1/x)*Dy+(1-1/x)*y=x^2*exp(-5*x)','y(1)=pi','y(pi)=1') y =

(exp(t)*(exp((x - 1)/x) - x*exp((x - 1)/x) - pi*exp((pi*(x - 1))/x) - (x^3*exp((pi*(x - 1))/x))/exp(5*x) + (x^3*exp((x - 1)/x))/exp(5*x) + pi*x*exp((pi*(x - 1))/x)))/(exp(pi)*exp((x - 1)/x) - exp(1)*exp((pi*(x - 1))/x) - x*exp(pi)*exp((x - 1)/x) + x*exp(1)*exp((pi*(x - 1))/x)) - (exp((t*(x - 1))/x)*(exp(1) - x*exp(1) - pi*exp(pi) - (x^3*exp(pi))/exp(5*x) + (x^3*exp(1))/exp(5*x) + pi*x*exp(pi)))/(exp(pi)*exp((x - 1)/x) - exp(1)*exp((pi*(x - 1))/x) - x*exp(pi)*exp((x - 1)/x) + x*exp(1)*exp((pi*(x - 1))/x)) + x^3/(exp(5*x)*(x - 1))

4、 Lotka-Volterra 扑食模型方程为()4()2()()

()()()3()

x t x t x t y t y t x t y t y t =-⎧⎨

=-⎩，且初值为(0)2,(0)3x y ==，

试求解该微分方程，并绘制相应的曲线。 >> syms x y t;

>> f=inline('[4*x(1)-2*x(1)*x(2); x(1)*x(2)-3*x(2)]','t','x'); >> [t,x]=ode45(f,[0,10],[2;3]);plot(t,x)