Q5Advanced Engineering Mathematics

The task requires approximating the solution to the non-linear first-order Ordinary Differential Equation (ODE) $\frac{dy}{dx}=x^2+y$, given the initial boundary condition $y(0)=1$, utilizing a precise step size of $h=0.02$. The Modified Euler's Method (also known as Heun's Method or a 2nd-order Runge-Kutta method) is a predictor-corrector algorithm that provides significantly better accuracy than the standard Forward Euler method by averaging the slopes at the beginning and end of each interval.

1. Iteration Step 1: Approximating y at x = 0.02

We commence at $x_0=0,y_0=1$. The function defines the slope: $f(x,y)=x^2+y$.

Predictor Phase: We use standard Euler to make an initial rough estimate ($y_1^{(0)}$) of the next point.

Corrector Phase (First Pass): We refine this estimate by averaging the slope at the initial point $(0,1)$ with the newly estimated slope at the predicted point $(0.02,1.02)$.

Corrector Phase (Second Pass): To ensure maximum numerical stability, we iteratively apply the corrector formula until the digits converge.

The value has essentially converged. We establish $y_1\approx 1.0202$ at $x_1=0.02$.

2. Iteration Step 2: Approximating y at x = 0.04

We now step forward to $x_2=0.04$, utilizing our newly established values $x_1=0.02$ and $y_1=1.0202$.

Predictor Phase:

Corrector Phase:

A secondary corrector pass would confirm convergence. Thus, the rigorously calculated approximation for $y(0.04)$ is approximately $1.0408$.