(a) Program a calculator or computer to use Euler's method...

(a) Program a calculator or computer to use Euler's method to compute $y(1),$ where $y(x)$ is the solution of the initial- value problem $$\frac{d y}{d x}+3 x^{2} y=6 x^{2} \quad y(0)=3$$ $$\begin{aligned} \text { (i) } h=1 & \text { (ii) } h=0.1 \\ \text { (iii) } h=0.01 & \text { (iv) } h=0.001 \end{aligned}$$ (b) Verify that $y=2+e^{-x^{3}}$ is the exact solution of the differential equation. (c) Find the errors in using Euler's method to compute $y(1)$ with the step sizes in part (a). What happens to the error when the step size is divided by 10$?$

(a) Program a calculator or computer to use Euler's method
to compute $y(1),$ where $y(x)$ is the solution of the initial-
value problem
$$\frac{d y}{d x}+3 x^{2} y=6 x^{2} \quad y(0)=3$$
$$\begin{aligned} \text { (i) } h=1 & \text { (ii) } h=0.1 \\ \text { (iii) } h=0.01 & \text { (iv) } h=0.001 \end{aligned}$$
(b) Verify that $y=2+e^{-x^{3}}$ is the exact solution of the
differential equation.
(c) Find the errors in using Euler's method to compute $y(1)$
with the step sizes in part (a). What happens to the error
when the step size is divided by 10$?$

Key Concepts

Error Analysis

Error analysis in the context of numerical differential equation solvers involves assessing the difference between the numerical approximation and the exact solution. In Euler's method, the local truncation error per step is generally proportional to h², while the global error over an interval is proportional to h. This analysis is important for understanding how reducing the step size (for instance, by a factor of 10) can lead to a significant decrease in the overall error, illustrating the convergence behavior of the method.

Euler's Method

Euler's method is a fundamental numerical technique used to approximate solutions to ordinary differential equations (ODEs). It works by advancing a solution step-by-step using the derivative’s value at the current point, effectively using a linear approximation over a small interval. This iterative process makes it a simple yet instructive method in numerical analysis for understanding how solutions can be built incrementally when an exact analytical solution may be difficult or impossible to obtain.

Initial Value Problem

An initial value problem (IVP) for an ordinary differential equation specifies the value of the unknown function at a given initial point, along with the differential equation that the function satisfies. By providing the initial condition, the IVP ensures a unique solution under typical conditions, and it forms the basis for applying numerical techniques like Euler's method to approximate the evolving behavior of the solution from the initial point.

Step Size

The step size, often denoted by h, is a key parameter in numerical methods such as Euler's method that dictates the interval between successive approximations. A smaller step size typically yields a more accurate approximation because it reduces the local error incurred in each step; however, it also requires more computational work. The relationship between step size and accuracy is crucial in balancing computational efficiency and the precision of the numerical solution.

Exact Solution Verification

Exact solution verification involves confirming that a proposed solution satisfies both the differential equation and the given initial condition. This is done by substituting the candidate solution into the differential equation and checking that the resulting equality holds for all relevant values. Such verification is crucial for validating the correctness of an exact analytical solution and serves as a benchmark for comparing the accuracy of numerical approximations.

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