Use Euler's Method with three equal step sizes to estimate the value of y(0.3) for the differential equation y ′ = y, with y(0) = 1.
Type your answer in the space below and give 3 decimal places. If your answer is less than 1, place a leading "0" before the decimal point (ex: 0.482).
To use Euler's method to estimate the value of y(0.3) for the differential equation y' = y, with y(0) = 1, we need to follow these steps:
Step 1: Determine the step size.
In this case, we are given that we should use three equal step sizes. So, we need to divide the interval [0, 0.3] into three equal parts. The step size, h, is calculated by dividing the width of the interval by the number of steps:
h = (0.3 - 0)/3 = 0.1
Step 2: Identify the number of steps.
As mentioned above, we need to perform three steps because we have three equal step sizes.
Step 3: Set up the Euler's method formula.
The Euler's method formula is given by:
y[n+1] = y[n] + h * f(x[n], y[n])
Step 4: Perform the calculations.
Let's start with the initial condition y(0) = 1. We have x[0] = 0 and y[0] = 1. Using these values and the formula from Step 3, we can calculate the approximations for y at each step.
Step 4.1: 1st step
x[1] = x[0] + h = 0 + 0.1 = 0.1
y[1] = y[0] + h * f(x[0], y[0])
= 1 + 0.1 * (y[0]) = 1 + 0.1 * 1 = 1.1
Step 4.2: 2nd step
x[2] = x[1] + h = 0.1 + 0.1 = 0.2
y[2] = y[1] + h * f(x[1], y[1])
= 1.1 + 0.1 * (y[1]) = 1.1 + 0.1 * 1.1 = 1.21
Step 4.3: 3rd step
x[3] = x[2] + h = 0.2 + 0.1 = 0.3
y[3] = y[2] + h * f(x[2], y[2])
= 1.21 + 0.1 * (y[2]) = 1.21 + 0.1 * 1.21 = 1.331
Step 5: Round and provide the answer.
The estimated value of y(0.3) using Euler's method with three equal step sizes is 1.331, rounded to 3 decimal places.
Tell you what. Read this and give it a shot.
http://tutorial.math.lamar.edu/Classes/DE/EulersMethod.aspx