Question

Use a numerical solver and Euler's method to obtain a four-decimal approximation of the indicated value.

y' = y − y², y(0) = 0.3;
y(0.5) ≈ (h = 0.1)
y(0.5) ≈ (h = 0.05)

Answer 1

To approximate y(0.5) for the differential equation y' = y - y² with y(0) = 0.3 using Euler's method, start with the initial point and follow an iterative process, calculating the slope at each step and estimating the next value of y using the chosen step size until reaching x = 0.5.

Step-by-step explanation:

The differential equation given is y' = y - y², with the initial condition y(0) = 0.3. To approximate the value of y(0.5) using Euler's method with step sizes h = 0.1 and h = 0.05, we will follow the iterative process of Euler's method.

1. Calculate the slope at the initial point using the differential equation.
2. Use this slope to estimate the value of y at the next step.
3. Repeat this process for each step until we reach x = 0.5.

Using a numerical solver or a calculator, we would input the step size and the differential equation to receive an output value, which would give us our approximations of y(0.5).

The examples given, such as starting with a value of y = 0.011 and refining it, or solving a quadratic equation, are illustrative of the iterative refinement process often used in numerical methods. While these specific values do not directly apply to our Euler's method calculation, they serve as examples of the process of iterative calculation and refinement in numerical analysis.