12.
WAP to implement Euler's Method for solving ordinary
differential equations (ODEs).
#include <iostream>
#include <iomanip>
#include <cmath>
using namespace std;
// Define the differential equation dy/dx = f(x, y)
double f(double x, double y) {
return x + y; // Example: dy/dx = x + y
}
int main() {
double x0, y0, xn, h;
int n;
// Input initial values
cout << "Enter the initial value of x (x0): ";
cin >> x0;
cout << "Enter the initial value of y (y0): ";
cin >> y0;
cout << "Enter the value of x at which to find y (xn): ";
cin >> xn;
cout << "Enter the number of steps (n): ";
cin >> n;
// Step size
h = (xn - x0) / n;
// Euler's Method
double x = x0;
double y = y0;
cout << fixed << setprecision(6);
cout << "\nStep\tx\ty" << endl;
for (int i = 0; i < n; ++i) {
cout << i + 1 << "\t" << x << "\t" << y << endl;
y = y + h * f(x, y); // Euler's formula
x = x + h;
}
// Final value
cout << "\nThe value of y at x = " << xn << " is approximately " << y << endl;
return 0;
}
Output:
13. WAP to implements a simple linear curve fitting method using
the least squares approach.
#include <iostream>
#include <vector>
#include <cmath>
#include <iomanip>
using namespace std;
// Function to calculate the sum of elements in a vector
double sum(const vector<double>& v) {
double total = 0;
for (double value : v) {
total += value;
}
return total;
}
// Function to calculate the sum of the product of corresponding elements of two vectors
double sum_of_products(const vector<double>& v1, const vector<double>& v2) {
double total = 0;
for (size_t i = 0; i < v1.size(); ++i) {
total += v1[i] * v2[i];
}
return total;
}
int main() {
int n;
cout << "Enter the number of data points: ";
cin >> n;
vector<double> x(n), y(n);
cout << "Enter the x values:\n";
for (int i = 0; i < n; ++i) {
cin >> x[i];
}
cout << "Enter the y values:\n";
for (int i = 0; i < n; ++i) {
cin >> y[i];
}
// Calculate required sums
double sum_x = sum(x);
double sum_y = sum(y);
double sum_x2 = sum_of_products(x, x);
double sum_xy = sum_of_products(x, y);
// Calculate coefficients a0 and a1 for y = a0 + a1 * x
double a1 = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x * sum_x);
double a0 = (sum_y - a1 * sum_x) / n;
// Display the equation of the fitted curve
cout << fixed << setprecision(6);
cout << "The equation of the fitted curve is: y = " << a0 << " + " << a1 << " * x" << endl;
return 0;
}
Output:
