Euler’s method Consider the initial value problem y′(t)=1/2y,y(0)=1. 
b. Use Euler’s method with Δt=0.05 to compute approximations to y(0.1) and y(0.2). 

Verified step by step guidance

1. Identify the differential equation and initial condition: \(y'(t) = \frac{1}{2} y\), with \(y(0) = 1\).
2. Recall Euler's method formula for approximating solutions: \(y_{n+1} = y_n + \Delta t \cdot f(t_n, y_n)\), where \(f(t, y) = y'(t)\).
3. Set the step size \(\Delta t = 0.05\) and start with \(t_0 = 0\), \(y_0 = 1\).
4. Calculate \(y_1\) at \(t_1 = 0.05\) using Euler's method: \(y_1 = y_0 + 0.05 \times \frac{1}{2} y_0\).
5. Calculate \(y_2\) at \(t_2 = 0.1\) similarly: \(y_2 = y_1 + 0.05 \times \frac{1}{2} y_1\). Repeat the process to find \(y_4\) at \(t_4 = 0.2\) by continuing the steps.

Key Concepts

Initial Value Problem (IVP)

An initial value problem involves a differential equation along with a specified value of the unknown function at a starting point. It defines a unique solution curve passing through the initial condition, allowing us to approximate or find the function's behavior over an interval.

Euler’s Method

Euler’s method is a numerical technique to approximate solutions of differential equations. It uses the slope at a known point to estimate the function’s value at the next step by moving forward in small increments (Δt), iteratively building an approximate solution.

Step Size (Δt) and Approximation Accuracy

The step size Δt determines the increments at which Euler’s method computes new values. Smaller Δt values generally yield more accurate approximations but require more computations, while larger Δt values increase error, affecting the precision of the solution.