//Eulers Method to solve a differential equation
#include<iostream>
#include<iomanip>
#include<cmath>
using namespace std;
double df(double x, double y) //function for defining dy/dx
{
double a=x+y; //dy/dx=x+y
return a;
}
int main()
{
int n;
double x0,y0,x,y,h; //for initial values, width, etc.
cout.precision(5); //for precision
cout.setf(ios::fixed);
cout<<"\nEnter the initial values of x and y respectively:\n"; //Initial values
cin>>x0>>y0;
cout<<"\nFor what value of x do you want to find the value of y\n";
cin>>x;
cout<<"\nEnter the width of the sub-interval:\n"; //input width
cin>>h;
cout<<"x"<<setw(19)<<"y"<<setw(19)<<"dy/dx"<<setw(16)<<"y_new\n";
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.
{
y=y0+(h*df(x0,y0)); //calculate new y, which is y0+h*dy/dx
cout<<x0<<setw(16)<<y0<<setw(16)<<df(x0,y0)<<setw(16)<<y<<endl;
y0=y; //pass this new y as y0 in the next iteration.
x0=x0+h; //calculate new x.
}
cout<<x0<<setw(16)<<y<<endl;
cout<<"The approximate value of y at x=0 is "<<y<<endl; //print the solution.
return 0;
}