常微分方程作業(yè)歐拉法與改進(jìn)歐拉法

1、P7731利用改進(jìn)歐拉方法計算下列初值問題，并畫出近似解的草圖:呻 y 1,y(0) 3,012't g代碼：液進(jìn)歐拉法fun ctio n Euler(tO,yO,i nv,h)n=rou nd(i nv(2)-i nv(1)/h;t(1)=tO;y(1)=y0;for i=1: ny1(i+1)=y(i)+h*fu n(t(i),y(i);t(i+1)=t(i)+h;y(i+1)=y(i)+1/2*h*(fu n(t(i),y(i)+ fun (t(i+1),y1(i+1)endplot(t,y,'*r')function y=fun( t,y);y=y+1;調(diào)用：

2、Euler(0,3,0,2, 得到解析解：hold on;y=dsolve('Dy=y+1','(y(0)=3)','t'); ezp lot(y,0,2)圖像:dy 2學(xué) y 4t,y(0)0.5,0 t 2, t 0.2;dt 1代碼：fun ctio n Euler1(t0,y0,i nv,h)n=rou nd(i nv(2)-i nv(1)/h;t(1)=t0;y(1)=y0;for i=1: ny1(i+1)=y(i)+h*fu n( t(i),y(i);t(i+1)=t(i)+h;y(i+1)=y(i)+1/2*h*(fu n(t(i

3、),y(i)+ fun (t(i+1),y1(i+1)end plot(t,y,'*門 function y=fun( t,y); y=yA2-4*t;調(diào)用:Euler1(0,0,2,圖像:-05-1.5-2-250.2040.6 O.B1.21.41.61.S2dy(3)£ (3 y)(y 1),y(0) 4,0 t 5, t 1; dt代碼：fun ctio n Euler2(t0,y0,i nv,h)n=rou nd(i nv(2)-i nv(1)/h;t(1)=t0;y(1)=y0;for i=1: ny1(i+1)=y(i)+h*fu n( t(i),y(i);t(

4、i+1)=t(i)+h;y(i+1)=y(i)+1/2*h*(fu n( t(i),y(i)+ fun( t(i+1),y1(i+1)endplot(t,y,'*r')function y=fun( t,y);y=(3-y)*(y+1);調(diào)用:Euler2(0,4,0,5,1)得到解析解：hold on;y=dsolve('Dy=(3-y)*(y+1)','y(0)=4','t'); ezpl ot(y)圖像：(A)lo-dz 寸H(0)A-(L+A)*(A-e)HACI-)e>_OSPHA-co P-OLI-_g-0寸-0

5、)22n山ML+A)*(Ag)HAXAeunE Uo4oun4(_*-A)10_dpu (L±)LA-(L±)l)un4+(M(m)uru)*LI*2/L+(_)AH(L±)A 二+(_)芥(L±)l gA-(m)un蕓+(_)ah(l±)laLrLU一046ah(l)a6芥(L)l w(L)>u_-(2)>u_)punoHU (LI->uroA-9)2_n山 UOQOUnAIP§-;寸(。)5 >)(；)刮寸)14.考慮滿足初始條件(x(0) , y(0) ) =(1,1)的下列方程組:dxdx(1):dty

6、,2x20.2y xy 1.2y ;選定時間步長t=，n=5.用改進(jìn)歐拉方法求兩個方程組的近似解;(1)代碼：fun ctio n Euler4(t0,i nt,n ,h)t=t0;x(1)=i nt(1);y(1)=in t(2);for i=1: nx1(i+1)=x(i)+h*xfu n(t(i),x(i),y(i);y1(i+1)=y(i)+h*yfu n(t(i),x(i),y(i);t(i+1)=t(i)+h;x(i+1)=x(i)+1/2*h*(xfu n( t(i),x(i),y(i)+xfu n(t(i+1),x1(i+1),y1(i+1); y(i+1)=y(i)+1/2*

7、h*(yfu n( t(i),x(i),y(i)+yfu n( t(i+1),x1(i+1),y1(i+1); endplot(t,x,'o-r')hold onplot(t,y,'*-g')hold onplot(x,y) function x=xfu n( t,x,y);x=y;function y=yfun( t,x,y);y=-2*x-3*y;調(diào)用函數(shù)：Euler4(0,1,1,5,圖像:(2)代碼：fun ctio n Euler5(t0,i nt,n,h) t=t0;x(1)=i nt(1);y(1)=in t(2);for i=1: nx1(i+1)=x(i)+h*xfu n(t(i),x(i),y(i);y1(i+1)=y(i)+h*yfu n( t(i),x(i),y(i);t(i+1)=t(i)+h;x(i+1)=x(i)+1/2*h*(xfu n( t(i),x(i),y(i)+xfu n(t(i+1),x1(i+1),y1(i+1); y(i+1)=y(i)+1/2*h*(yfu n( t(i),x(i),y(i)+yfu n( t(i+1),x1(i+1),y1(i+1); endplot(t,x,'o-r')hold onplot(t,y,'