1. The improved Euler method, also known as Heun's method, is a numerical technique to solve ordinary differential equations (ODEs) using an improved slope estimate. 2. Given an initial value problem $$\frac{dy}{dx} = f(x,y)$$ with initial condition $$y(x_0)=y_0$$, the goal is to find $$y$$ at $$x_{n+1} = x_n + h$$ where $$h$$ is the step size. 3. The improved Euler formula computes the next value $$y_{n+1}$$ by averaging slopes at the beginning and predicted end of the interval: $$k_1 = f(x_n, y_n)$$ $$k_2 = f(x_n + h, y_n + h k_1)$$ Then, $$y_{n+1} = y_n + \frac{h}{2}(k_1 + k_2)$$ 4. To prove this formula, consider the exact solution expanded by Taylor series about $$x_n$$: $$y(x_n + h) = y(x_n) + h y'(x_n) + \frac{h^2}{2} y''(x_n) + O(h^3)$$ Using $$y' = f(x,y)$$ and chain rule for the second derivative, the improved Euler method matches the Taylor expansion terms up to order $$h^2$$, showing it is a second order method. 5. Thus, improved Euler's method gives a better slope estimate than the basic Euler method which uses only $$k_1$$, because it incorporates the slope at the predicted point $$k_2$$, reducing local truncation error. Final formula: $$\boxed{y_{n+1} = y_n + \frac{h}{2} \left( f(x_n,y_n) + f(x_n + h, y_n + h f(x_n,y_n)) \right)}$$