Use Euler's method with step size .2 to estimate y(.4), where y(x) is the solution of the initial value problem y=x+y^2, y=0.
Repeat part a with step size .1
To use Euler's method to estimate y(0.4) for the initial value problem y=x+y^2, y=0, we first need to find the values of y at different increments of x.
Step 1: Determine the number of steps.
Given a step size of 0.2, the number of steps required to reach x = 0.4 is 0.4 / 0.2 = 2.
Step 2: Initialize variables.
Set x0 = 0 and y0 = 0, which are the initial values.
Step 3: Calculate the estimated values.
Using Euler's method, calculate the estimated values of y for each step.
For the first step:
x1 = x0 + h = 0 + 0.2 = 0.2
y1 = y0 + h * (dy/dx) = y0 + 0.2 * (x0 + y0^2) = 0 + 0.2 * (0 + 0^2) = 0 + 0 = 0
For the second step:
x2 = x1 + h = 0.2 + 0.2 = 0.4
y2 = y1 + h * (dy/dx) = y1 + 0.2 * (x1 + y1^2) = 0 + 0.2 * (0.2 + 0^2) = 0 + 0.2 * 0.2 = 0.04
Therefore, the estimated value of y(0.4) using Euler's method with a step size of 0.2 is 0.04.
To repeat the process with a step size of 0.1, we need to calculate the values at smaller increments.
Step 1: Determine the number of steps.
Given a step size of 0.1, the number of steps required to reach x = 0.4 is 0.4 / 0.1 = 4.
Step 2: Initialize variables.
Set x0 = 0 and y0 = 0, which are the initial values.
Step 3: Calculate the estimated values.
Using Euler's method, calculate the estimated values of y for each step.
For the first step:
x1 = x0 + h = 0 + 0.1 = 0.1
y1 = y0 + h * (dy/dx) = y0 + 0.1 * (x0 + y0^2) = 0 + 0.1 * (0 + 0^2) = 0 + 0 = 0
For the second step:
x2 = x1 + h = 0.1 + 0.1 = 0.2
y2 = y1 + h * (dy/dx) = y1 + 0.1 * (x1 + y1^2) = 0 + 0.1 * (0.1 + 0^2) = 0 + 0.1 * 0.1 = 0.01
For the third step:
x3 = x2 + h = 0.2 + 0.1 = 0.3
y3 = y2 + h * (dy/dx) = y2 + 0.1 * (x2 + y2^2) = 0.01 + 0.1 * (0.2 + 0.1^2) = 0.01 + 0.1 * 0.21 = 0.01 + 0.021 = 0.031
For the fourth step:
x4 = x3 + h = 0.3 + 0.1 = 0.4
y4 = y3 + h * (dy/dx) = y3 + 0.1 * (x3 + y3^2) = 0.031 + 0.1 * (0.3 + 0.031^2) = 0.031 + 0.1 * 0.307961 = 0.031 + 0.0307961 = 0.0617961
Therefore, the estimated value of y(0.4) using Euler's method with a step size of 0.1 is 0.0617961.
To use Euler's method, we need to first determine the number of steps and the step size. Given the initial value problem y = x + y^2 and y = 0, we want to estimate y(0.4).
a) Using a step size of 0.2:
Step 1: Determine the number of steps.
Since the step size is 0.2, the number of steps required to reach x = 0.4 is 0.4 / 0.2 = 2 steps.
Step 2: Perform the iterative calculations.
Let's denote x_i as the current x-value and y_i as the corresponding estimated y-value.
For the first step (i = 1):
x_1 = 0 + 0.2 = 0.2
y_1 = 0 + (0.2)^2 = 0.04
For the second step (i = 2):
x_2 = 0.2 + 0.2 = 0.4
y_2 = 0.2 + (0.04)^2 = 0.20016
Therefore, the estimated value of y(0.4) using Euler's method with a step size of 0.2 is y(0.4) ≈ y_2 = 0.20016.
b) Using a step size of 0.1:
Step 1: Determine the number of steps.
Since the step size is 0.1, the number of steps required to reach x = 0.4 = 0.4 / 0.1 = 4 steps.
Step 2: Perform the iterative calculations.
For the first step (i = 1):
x_1 = 0 + 0.1 = 0.1
y_1 = 0 + (0.1)^2 = 0.01
For the second step (i = 2):
x_2 = 0.1 + 0.1 = 0.2
y_2 = 0.1 + (0.01)^2 = 0.1001
For the third step (i = 3):
x_3 = 0.2 + 0.1 = 0.3
y_3 = 0.2 + (0.1001)^2 = 0.20100301
For the fourth step (i = 4):
x_4 = 0.3 + 0.1 = 0.4
y_4 = 0.3 + (0.20100301)^2 ≈ 0.301207029
Therefore, the estimated value of y(0.4) using Euler's method with a step size of 0.1 is y(0.4) ≈ y_4 ≈ 0.301207029.