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)=0
Estimate y(1.4) with a step size h=0.2
y(1.4)=_____

Answer 1

To estimate y(1.4) using Euler's method, find the derivative of y and calculate the slopes at each step using the given step size. Use these slopes to approximate the value of y at x=1.4.

Step-by-step explanation:

To estimate y(1.4) using Euler's method, we will first find the slope at the initial point, which is given by the derivative of y. The derivative of y is y′=x-xy. Plugging in the initial values, we get y′=1-(1)(0)=1. Now, we will use this slope to approximate the value of y at x=1.2. Using the step size of h=0.2, we calculate y(1.2) as follows:

y(1.2) = y(1) + (h * y′) = 0 + (0.2 * 1) = 0.2

Next, we will use the newly estimated value of y(1.2) to calculate the slope at x=1.2, which gives us y′=1.2-1.2(0.2)=0.96. Finally, using this slope, we can find the approximation of y(1.4):

y(1.4) = y(1.2) + (h * y′) = 0.2 + (0.2 * 0.96) = 0.392