Question

It can be shown that the solution to the initial value problem:

y' = e^(-π³/³), y(1) = 0

is:

y(x) = ∫ from 1 to x of e^(-t²) dt

Use Euler's Method with a step size of Δx = 0.2 to approximate y(1.6). Start with the initial condition y(1) = 0.

Answer 1

The question asks to approximate y(1.6) for a given initial value problem using Euler's Method with a step size of 0.2. Euler's Method is a numerical technique for solving differential equations, which in this case requires iteratively calculating the value of y at successive steps.

Step-by-step explanation:

The question involves applying Euler's Method to approximate the value of a function at a given point, based on a differential equation and an initial value. Euler's Method is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.

To approximate y(1.6) using Euler's method with a step size of Δx = 0.2, we start at the initial condition y(1) = 0 and use the derivative y' = e^(-π³/3) to calculate the values of y at subsequent steps. The steps involved are:

• Calculate y(1.2) using the initial value and derivative.
• Calculate y(1.4) using the value from the previous step and the derivative.
• Finally, calculate y(1.6) using the value from the last step and the derivative.

By following these steps, we can approximate y(1.6) without directly integrating the function.