Trying to solve this Second Order ODE in both excel and C++ using Heun\'s Method

Question

Trying to solve this Second Order ODE in both excel and C++ using Heun's Method:

Write a C++ program and excel spreadsheet using the shooting method in combination with Heun's Method that can solve the following Boundary-Value Second Order Differential Equation: y'' + 4y' + 4y = 0, with y(0)=0 and y(1)=3.

The program must prompt the user to provide a first guess of the slope at x = 0. You can select any step size you consider reasonable; proof of solution will be decided on how well the calculated results match the analytical solution at x=1.

The analytical solution is: y = 3xe^(-2x+2)

Explanation / Answer

#include <cstdlib>
#include <iostream>
#include <cmath>
#include <iomanip>
using namespace std;
long double f(long double x11, long double y11) {
    return ((3 * exp(0.15*x11)) - (0.15*y11));
}

long double g(long double x11) {
    return (4/1.3)*(exp(0.15*x11)-exp(-0.15*x11))+2*exp(-0.15*x11);
}

long double hf0(long double x12, long double y12, long double h1) {
    return y12 + (f(x12, y12) * h1);
}

long double hf0_corrector(long double x12, long double y12, long double x21, long double y22, long double h1) {
    return y12 + (((f(x12, y12) + f(x21, y22)) / 2)*h1);
}

long double get_error(long double a11, long double b11) {
    return abs(((a11-b11)*100/a11));
}

int main(int argc, char** argv) {
    cout << fixed;
    cout << setprecision (60);

    long double x01 = 0;
    long double y01 = 2;
    long double step = 1e-3;
    long double x_mx = 4;

    long double pred = 0;
    long double targeterror = 2e-5;
    long double err = 100;
    long double correctornew = 0;
    unsigned long int i11 = 0;
    long double ytrue;
    long double etrue = 100;
    long double etrue_now = 100;
    long double corre;

    long double true_maxerror = 0;

    for (x01= 0; x01<=x_mx; x01=x01+step) {
        targeterror = 2e-5;
        err = 100;
        etrue = 100;
        etrue_now = 101;
        correctornew = 0;
        i11 = 0;

        ytrue = g(x01 + step);
        pred = hf0(x01, y01, step);
        corre = hf0_corrector(x01, y01, (x01 + step), pred, step);
      
        while (etrue > 1) {
            correctornew = hf0_corrector(x01, y01, (x01 + step), corre, step);
            err = get_error(corre, correctornew);
            etrue_now = get_error(corre, ytrue);
            corre = correctornew;
            i11++;

       
            if (etrue_now == etrue) break;

            etrue = etrue_now;
        }

        if (true_maxerror < etrue) {
            true_maxerror = etrue;
        }
        y01= corre;
    }

    cout << "at f(" << x01 << "): " << endl;
    cout << "estd y: " << corre << endl;
  

    return 0;
}

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