1.1、求解初值问题()⎪⎩⎪⎨⎧=-=-10y yxe dx dyx ，已知精确解为()()x x x x y -+=2212当h=0.1时，解为：n xn y()n x y()nn y x y -0 1 1 00.1 0.900000 0.909362 9.3616E-03 0.2 0.819048 0.835105 1.6057E-02 0.3 0.753518 0.774155 2.0637E-02 0.4 0.700391 0.723946 2.3555E-02 0.5 0.657165 0.682347 2.5182E-02 0.6 0.621775 0.647598 2.5823E-02 0.7 0.592526 0.618249 2.5723E-02 0.8 0.568034 0.593114 2.5080E-02 0.9 0.547177 0.571230 2.4053E-02 1.00.529051 0.551819 2.2768E-020.10.20.30.40.50.60.70.80.91当h=0.05时，解为：n xn y()n x y()nn y x y -0 1 1 00.05 0.950000 0.952418 2.4185E-03 0.10 0.904878 0.909362 4.4835E-03 0.15 0.864158 0.870391 6.2326E-03 0.20 0.827406 0.835105 7.6996E-03 0.25 0.794223 0.803138 8.9155E-03 0.30 0.764247 0.774155 9.9084E-03 0.35 0.737147 0.747850 1.0704E-02 0.40 0.712621 0.723946 1.1324E-02 0.45 0.690397 0.702188 1.1791E-02 0.50 0.670223 0.682347 1.2124E-02 0.55 0.651876 0.664213 1.2338E-02 0.60 0.635148 0.647598 1.2450E-02 0.65 0.619855 0.632328 1.2473E-02 0.70 0.605829 0.618249 1.2420E-02 0.75 0.592918 0.605220 1.2302E-02 0.80 0.580985 0.593114 1.2129E-02 0.85 0.569909 0.581819 1.1909E-02 0.90 0.559579 0.571230 1.1651E-02 0.95 0.549896 0.561258 1.1362E-02 1.000.5407710.5518191.1048E-020.10.20.30.40.50.60.70.80.91h=50时，解为：n xn y()n x y()n n y x y -0 1 1 00.02 0.980000 0.980395 3.9471E-04 0.04 0.960792 0.961558 7.6599E-04 0.06 0.942345 0.943460 1.1148E-03 0.08 0.924628 0.926070 1.4422E-03 0.10 0.907613 0.909362 1.7491E-03 0.12 0.891270 0.893306 2.0363E-03 0.14 0.875573 0.877878 2.3048E-03 0.16 0.860496 0.863051 2.5553E-03 0.18 0.846013 0.848802 2.7888E-03 0.20 0.832100 0.835105 3.0058E-03 0.22 0.818732 0.821940 3.2073E-03 0.24 0.805889 0.809283 3.3938E-03 0.26 0.793547 0.797113 3.5662E-03 0.28 0.781685 0.785410 3.7250E-03 0.30 0.770284 0.774155 3.8709E-03 0.32 0.759323 0.763328 4.0045E-03 0.34 0.748784 0.752911 4.1264E-03 0.36 0.738649 0.742886 4.2371E-03 0.38 0.728899 0.733236 4.3373E-03 0.40 0.719518 0.723946 4.4274E-03 0.42 0.710490 0.714998 4.5079E-03 0.44 0.701800 0.706379 4.5793E-03 0.46 0.693431 0.698073 4.6421E-03 0.48 0.685371 0.690067 4.6967E-03 0.50 0.677603 0.682347 4.7435E-03 0.52 0.670117 0.674900 4.7830E-03 0.54 0.662897 0.667713 4.8156E-03 0.56 0.655933 0.660775 4.8415E-03 0.58 0.649212 0.654073 4.8613E-03 0.60 0.642723 0.647598 4.8751E-03 0.62 0.636454 0.641337 4.8835E-03 0.64 0.630395 0.635282 4.8866E-03 0.66 0.624537 0.629422 4.8848E-03 0.68 0.618868 0.623747 4.8784E-03 0.70 0.613381 0.618249 4.8676E-03 0.72 0.608066 0.612918 4.8528E-03 0.74 0.602914 0.607748 4.8341E-03 0.760.5979170.6027284.8119E-030.78 0.593067 0.597853 4.7863E-03 0.80 0.588357 0.593114 4.7577E-03 0.82 0.583779 0.588505 4.7261E-03 0.84 0.579326 0.584018 4.6918E-03 0.86 0.574992 0.579647 4.6550E-03 0.88 0.570771 0.575387 4.6159E-03 0.90 0.566656 0.571230 4.5746E-03 0.92 0.562641 0.567172 4.5314E-03 0.94 0.558721 0.563207 4.4864E-03 0.96 0.554890 0.559330 4.4397E-03 0.98 0.551144 0.555535 4.3916E-03 1.000.547477 0.551819 4.3420E-030.10.20.30.40.50.60.70.80.91有图像看出，当步长越小，计算得到的解越逼近精确解。

1.2、求解初值问题⎪⎩⎪⎨⎧=++-==110x y x y dx dy，已知精确解为：x e x y -+=，h=0.1欧拉法图像：改进欧拉法图像：预测-校正法图像：附录：源代码1.1、clearclcX0=0;X1=1;n=10;%更改分点数h=1/n;%步长y(1)=1;x(1)=X0;for i=1:nx(i+1)=x(i)+h;y(i+1)=y(i)+h*(x(i)*exp(-x(i))-y(i)); endx=vpa(x',6)y=vpa(y',6)X=(X0:0.001:X1);n=1/0.001;for(i=1:n+1)Y(i)=0.5*(X(i)^2+2)*exp(-X(i));endplot(x,y,'*')hold onplot(X,Y)X=X0:h:X1;Y=0.5.*(X.^2+2).*exp(-X);Y=vpa(Y',6)yy=abs(y-Y)1.2、欧拉法：clearclcX0=0;X1=1;n=10;h=1/n;y(1)=1;x(1)=X0;X=X0:h:X1;Y=X+exp(-X);Y=vpa(Y',9);%精确解XX=X0:0.0001:X1;YY=XX+exp(-XX);for i=1:nx(i+1)=x(i)+h;y(i+1)=y(i)+h*(-y(i)+x(i)+1); endy=vpa(y',9)plot(x,y,'*')hold onplot(XX,YY)改进的欧拉法：clearclcX0=0;X1=1;n=10;h=1/n;y(1)=1;x(1)=X0;X=X0:h:X1;Y=X+exp(-X);Y=vpa(Y',9);%精确解XX=X0:0.0001:X1;YY=XX+exp(-XX);for i=1:nx(i+1)=x(i)+h;y(i+1)=y(i)+h*(-y(i)+x(i)+1);endfor i=1:ny(i+1)=y(i)+0.5*h*((-y(i)+x(i)+1)+(-y(i+1)+x(i+1)+1));endy=vpa(y',9)plot(x,y,'*')hold onplot(XX,YY)预报-校正格式：clearclcX0=0;X1=1;n=10;h=1/n;y(1)=1;x(1)=X0;X=X0:h:X1;Y=X+exp(-X);Y=vpa(Y',9);%精确解XX=X0:0.0001:X1;YY=XX+exp(-XX);for i=1:nx(i+1)=x(i)+h;y(i+1)=y(i)+h*(-y(i)+x(i)+1);endfor i=1:nfor count=1:10 %预报-校正格式，迭代十次y(i+1)=y(i)+0.5*h*((-y(i)+x(i)+1)+(-y(i+1)+x(i+1)+1));endendy=vpa(y',9)plot(x,y,'*')hold onplot(XX,YY)。