Hello! Just needed help with understanding this specific question. We never really wet over Euler's method so I'm not sure how to go about it.
"Use Euler's method in order to solve the initial value problem below.
dy/dx = x-3 and y=4 when x=1
Use Euler's method with increment's of (triangle symbol)x=0.1 to approximate the value of y when x=1.3
A. 3.48
B. 3.68
C. 3.5
D. 3.43"
Thank you so much for your help.
y' = x-3
P0 = (1,4)
∆y = y'(x0)*∆x = (-2)(0.1) = -0.2
P1 = (1.1,3.8)
∆y = y'(x1)*∆x = (-1.9)(0.1) = -0.19
P2 = (1.2,3.61)
∆y = y'(x2)*∆x = (-1.8)(0.1) = -0.18
P3 = (1.3,3.43)
Read the fine article at wikipedia, which shows a geometric interpretation of the method.
To solve this problem using Euler's method, you need to follow these steps:
Step 1: Determine the step size (triangle symbol)x.
The problem states that the step size (triangle symbol)x is equal to 0.1. This means that for each step, the value of x will increase by 0.1.
Step 2: Determine the number of steps.
To find the number of steps required to reach x = 1.3, we need to subtract the initial value of x (1.0) from the target value (1.3) and divide it by the step size:
Number of steps = (1.3 - 1.0) / 0.1 = 3
Therefore, we need to perform the Euler's method for 3 steps.
Step 3: Perform the Euler's method.
- Start with the initial value given: y = 4 when x = 1.
- Calculate dy/dx = x - 3 for the given value (x = 1).
dy/dx = 1 - 3 = -2
- Calculate the next value of y using Euler's method:
y1 = y0 + dy/dx * (triangle symbol)x
For each step from 1 to 3, calculate the new value of y:
Step 1:
- x1 = 1 + (0.1) = 1.1
- dy/dx at x1 = x1 - 3 = 1.1 - 3 = -1.9
- y1 = 4 + (-1.9) * (0.1) = 4 - 0.19 = 3.81
Step 2:
- x2 = 1.1 + (0.1) = 1.2
- dy/dx at x2 = x2 - 3 = 1.2 - 3 = -1.8
- y2 = 3.81 + (-1.8) * (0.1) = 3.81 - 0.18 = 3.63
Step 3:
- x3 = 1.2 + (0.1) = 1.3
- dy/dx at x3 = x3 - 3 = 1.3 - 3 = -1.7
- y3 = 3.63 + (-1.7) * (0.1) = 3.63 - 0.17 = 3.46
Step 4:
We have reached the desired value of x (x = 1.3) using three steps.
The approximate value of y at x = 1.3 is 3.46.
Therefore, the correct answer is D. 3.43.
Keep in mind that Euler's method is an approximation method and the accuracy of the result depends on the step size used. Smaller step sizes generally result in more accurate approximations.