Question

Use Euler's method with the given step size to estimate y(1.4) where y(x) is the solution of the initial-value problem y′=x−xy ,y(1)=2. 1. Estimate y(1.4) with a step size h=0.2. Answer: y(1.4)≈ 2. Estimate y(1.4) with a step size h=0.1. Answer: y(1.4)≈

Answer 1

To estimate y(1.4) using Euler's method, we can calculate the values of y(1.2) and y(1.4) with a step size of 0.2.

Step-by-step explanation:

In order to estimate the value of y(1.4) using Euler's method, we need to use the given step size h and the initial value y(1) = 2. Let's first calculate the value of y(1.2) with a step size of 0.2:

y(1.2) = y(1) + h * f(1, y(1))

where f(x, y) = x - x * y. Plugging in the values, we get:

y(1.2) = 2 + 0.2 * (1 - 1 * 2) = 2 + 0.2 * (-1) = 2 - 0.2 = 1.8

Similarly, we can calculate the value of y(1.4) using a step size of 0.2:

y(1.4) = y(1.2) + h * f(1.2, y(1.2))

Plugging in the values, we get:

y(1.4) = 1.8 + 0.2 * (1.2 - 1.2 * 1.8) = 1.8 + 0.2 * (1.2 - 3.36) = 1.8 + 0.2 * (-2.16) = 1.8 - 0.432 = 1.368