Run the code online here:

https://onlinegdb.com/N-6KusNbR

Code:

//Modified Euler's Method for differential equations
#include<iostream>
#include<math.h>
#include<iomanip>
using namespace std;
double df(double x, double y)
{
    double a=x+y;            //function for defining dy/dx
    return a;
}
int main()
{
    double x0,y0,x,y_i,dy1,dy2,dy_avg,y_n,h;    //for initial values, width, etc.
    cout<<"\nEnter the initial values of x and y respectively:\n"; cin>>x0>>y0;                    //Initial values
    cout<<"\nEnter the value of x for which you would like to find y:\n"; cin>>x;
    cout<<"\nEnter the interval width,h:\n"; cin>>h;                        //input width
    cout<<"x"<<setw(16)<<"y"<<setw(16)<<"hy'"<<setw(16)<<"y_new"<<setw(16)<<"hy_new'"<<setw(16)<<"hy'avg"<<setw(16)<<"y_n+1"<<endl;
    cout<<"--------------------------------------------------------------------------------------------------\n"; 
    while(fabs(x-x0)>0.0000001)      //I couldn't just write "while(x0<x)" as they both are floating point nos. It is dangerous to compare two floating point nos. as they are not the same in binary as they are in decimal. For instance, a computer cannot exactly represent 0.1 or 0.7 in binary just like decimal can't represent 1/3 exactly without recurring digits.
    {   
        dy1=h*df(x0,y0);        //calculate slope or dy/dx at x0,y0
        y_i=y0+dy1;            //calculate new y, which is y0+h*dy/dx
        dy2=h*df(x0+h,y_i);        //calculate slope or dy/dx at x0+h (new x),new y
        dy_avg=(dy1+dy2)/2.0;        //calculate the average of the slopes at y0 and new y
        y_n=y0+dy_avg;            //calculate new y, which is y0+h*average(dy/dx)   
        cout<<x0<<setw(16)<<y0<<setw(16)<<dy1<<setw(16)<<y_i<<setw(16)<<dy2<<setw(16)<<dy_avg<<setw(16)<<y_n<<endl;
        x0=x0+h;            //calculate new x.
        y0=y_n;                //pass this new y as y0 in the next iteration.
    }
    cout<<x0<<setw(16)<<y0<<endl;
    cout<<"The approximate value of y at x=0 is "<<y0<<endl;    //print the solution.
    return 0;
}

Run the code online here:

https://onlinegdb.com/N-6KusNbR

Sample output

Exact (ideal) solution

Explaination of the code:

Manas Sharma

I’m a physicist specializing in computational material science with a PhD in Physics from Friedrich-Schiller University Jena, Germany. I write efficient codes for simulating light-matter interactions at atomic scales. I like to develop Physics, DFT, and Machine Learning related apps and software from time to time. Can code in most of the popular languages. I like to share my knowledge in Physics and applications using this Blog and a YouTube channel.

manas.bragitoff.com/