修改下面的代码 #include <iostream> #include<math.h> #include<iomanip> using namespace std; double f(double y,double x) { return y-2*x/y; } double four(double h,double y0,double x0,int N) { int n=1; double x1,k1,k2,k3,k4,y1; cout<<"四阶龙格-库塔法:"<<endl; while(n!=N+1) { x1=x0+h; k1=f(y0,x0); k2=f(y0+h*k1/2,x0+h/2); k3=f(y0+h*k2/2,x0+h/2); k4=f(y0+h*k3,x1); y1=y0+(k1+2*k2+2*k3+k4)*h/6; cout << setiosflags(ios::right) << setw(12) <<setiosflags(ios::fixed)<<setprecision(2)<<x1<<"|"<< setiosflags(ios::right) << setw(12) <<setiosflags(ios::fixed)<<setprecision(8)<< y1; cout<<"|"<< setiosflags(ios::right) << setw(12) <<setiosflags(ios::fixed)<<setprecision(8)<< sqrt(1+2*x1)<<endl; n+=1; x0=x1; y0=y1; } cout<<"------------------------------------"<<endl; } int main() { cout<<"输入步长h："; double h; cin>>h; cout<<"输入步数N："; int N; cin>>N; double y0=1,x0=0; Euler(h,y0,x0,N); beuler(h,y0,x0,N); four(h,y0,x0,N); return 0; }

根据代码中的函数名和调用，可以看出代码中缺少 Euler 和 beuler 函数的定义，因此需要添加这两个函数的定义。同时，函数 four 的调用中应该修改为 Euler 函数的调用。修改后的代码如下：

#include <iostream> 
#include <math.h> 
#include <iomanip> 

using namespace std; 

double f(double y, double x) {
    return y - 2*x/y;
}

double Euler(double h, double y0, double x0, int N) {
    int n = 1;
    double x1, y1;
    cout << "欧拉法：" << endl;
    while (n != N+1) {
        x1 = x0 + h;
        y1 = y0 + h * f(y0, x0);
        cout << setiosflags(ios::right) << setw(12) << setiosflags(ios::fixed) << setprecision(2) << x1 << "|"
             << setiosflags(ios::right) << setw(12) << setiosflags(ios::fixed) << setprecision(8) << y1;
        cout << "|" << setiosflags(ios::right) << setw(12) << setiosflags(ios::fixed) << setprecision(8)
             << sqrt(1+2*x1) << endl;
        n += 1;
        x0 = x1;
        y0 = y1;
    }
    cout << "------------------------------------" << endl;
}

double beuler(double h, double y0, double x0, int N) {
    int n = 1;
    double x1, y1, y_temp;
    cout << "改进欧拉法：" << endl;
    while (n != N+1) {
        x1 = x0 + h;
        y_temp = y0 + h * f(y0, x0);
        y1 = y0 + h / 2 * (f(y0, x0) + f(y_temp, x1));
        cout << setiosflags(ios::right) << setw(12) << setiosflags(ios::fixed) << setprecision(2) << x1 << "|"
             << setiosflags(ios::right) << setw(12) << setiosflags(ios::fixed) << setprecision(8) << y1;
        cout << "|" << setiosflags(ios::right) << setw(12) << setiosflags(ios::fixed) << setprecision(8)
             << sqrt(1+2*x1) << endl;
        n += 1;
        x0 = x1;
        y0 = y1;
    }
    cout << "------------------------------------" << endl;
}

double four(double h, double y0, double x0, int N) {
    int n = 1;
    double x1, k1, k2, k3, k4, y1;
    cout << "四阶龙格-库塔法：" << endl;
    while (n != N+1) {
        x1 = x0 + h;
        k1 = f(y0, x0);
        k2 = f(y0 + h*k1/2, x0+h/2);
        k3 = f(y0 + h*k2/2, x0+h/2);
        k4 = f(y0 + h*k3, x1);
        y1 = y0 + (k1 + 2*k2 + 2*k3 + k4) * h / 6;
        cout << setiosflags(ios::right) << setw(12) << setiosflags(ios::fixed) << setprecision(2) << x1 << "|"
             << setiosflags(ios::right) << setw(12) << setiosflags(ios::fixed) << setprecision(8) << y1;
        cout << "|" << setiosflags(ios::right) << setw(12) << setiosflags(ios::fixed) << setprecision(8)
             << sqrt(1+2*x1) << endl;
        n += 1;
        x0 = x1;
        y0 = y1;
    }
    cout << "------------------------------------" << endl;
}

int main() {
    cout << "输入步长h：";
    double h;
    cin >> h;
    cout << "输入步数N：";
    int N;
    cin >> N;
    double y0 = 1, x0 = 0;
    Euler(h, y0, x0, N);
    beuler(h, y0, x0, N);
    four(h, y0, x0, N);
    return 0;
}