A student is using Euler's method to solve a differential equation from $x = 0$ to $x = 10$. They initially choose a step size of $h = 0.1$. To improve accuracy, they decide to halve the step size to $h = 0.05$.

What is the expected effect on the local truncation error per step, and what is the expected effect on the overall global truncation error at $x = 10$?

For a certain application, a global error of no more than $10^{-4}$ is acceptable. Using Euler's method, a step size of $h=10^{-4}$ is required. If the user switches to the 4th-order Runge-Kutta method (RK4), approximately what step size $h$ would yield the same acceptable global error?

Given the initial value problem:

Use Euler's method with a step size $h = 0.1$ to approximate $y(0.2)$.

Approximate $y(1.2)$ for the IVP $\frac{dy}{dx} = xy^2$, $y(1) = 2$ using Euler's Method with $h = 0.1$.

Using the same IVP $\frac{dy}{dx} = x + y, y(0) = 1$, use Improved Euler's method with $h = 0.1$ to approximate $y(0.1)$.

Approximate $y(1.1)$ for $\frac{dy}{dx} = xy^2$, $y(1) = 2$ using Improved Euler with $h = 0.1$.

For the IVP $\frac{dy}{dx} = x^2 - y, \quad y(0) = 1$, find the first step using RK4 with $h=0.1$.

Approximate $y(0.1)$ for $\frac{dy}{dx} = 2y$, $y(0)=3$ using RK4 with $h=0.1$.