//**************************************************************************
//function is f(x)=0.2xy
//f(1)=1.
//h = b-a/n,  (1.0-1.5)/100
//Programmer:  David R. Soto,  
//Dated: April 15,  IRS day,  1995
//Approximation of f(1.5) for the solution of f'(x)=0.2xy, f(1)=1
//Euler method using h=0.1,  k.a. method of tangent lines.
//*************************************************************************************************************
//#include <iostream.h>
#include <stdio.h>		//Preprocessor directive for cout output

int n=500;                 				//we can change the width of the intervals easier by placing the
																	// n so that we have global scope.
double a=1.0, b=1.5;  						// the with of our x-axis will be 0.5.
double x1 = 1.0;         					//here,  we declare and initialize X.
double y1 = 1.0;         					// This is the initial value of the function.

double func(double x, double y);//This is a function prototype,  It is declared before the main
				// and used below.  The procedure follows the main.

//************************************the program enters main()****************************************
main()
{ 

	float h;												//Declaration of local variable of       								//type float used to hold the width h.
	h=(b-a)/n;
printf("\n\nDemonstration of Euler method,  used to calculate the approximation to some Func\n");
printf("the function is now changed to f(x)=0.2xy,  and f(1)=1.\n");
printf("The interval size is:%16.10f\n", h); 
//	cout<<"\n\nDemonstration of Euler method,  used to calculate the approximation to\n";
//	cout<<"the function is now changed to f(x)=0.2xy,  and f(1)=1.\n";
//	cout<< "The interval size is:" << h <<endl;
	for  (x1 = 1.0;  x1 <=1.5;  x1+h)
	{
	y1 = y1 + h*func(x1,y1);				//main invokes Aux function
	x1= x1+h;												//Updating x by adding another interval
	}
	printf("approx val of f at :%16.10f\n", x1);
	printf("is : %16.10f\n" , y1);
//	cout<<"Approximate value of the function at " <<x1<< ": " <<y1<<endl;
//	cout <<"User can change the size of the interval and observe results when h is changed\n";
return 0;
}

//*******************************the auxiliary function 
double func(double x, double y)
{

	y = (0.2*x*y);
	return y;
}