All Flashcards
How do you set up the table for Euler's Method?
Columns: x_i, y_i, f(x_i, y_i), Δy, y_{i+1}. Fill initial condition, calculate f(x_i, y_i), then Δy, and y_{i+1}.
Given a differential equation and initial condition, how do you find y(a) using Euler's method?
Determine step size h. Iteratively calculate x_{i+1}, y_{i+1} until x_{i+1} is close to a. The final y_{i+1} is the approximation of y(a).
How do you determine the step size if you're given a range and number of steps?
Calculate h = (x_final - x_initial) / number of steps.
How do you calculate the slope at each step in Euler's method?
Substitute the current x and y values into the differential equation dy/dx = f(x, y).
How do you find the next x-value in the Euler's method iteration?
Add the step size to the current x-value: x_{i+1} = x_i + h.
What are the first steps in applying Euler's method?
Identify the differential equation, initial condition, and step size.
How do you handle a differential equation that is not in the form dy/dx = f(x, y)?
Rearrange the equation to isolate dy/dx on one side.
How do you estimate y(2) using Euler's method with a step size of 0.5, given y' = x + y and y(0) = 1?
Iterate: x_0 = 0, y_0 = 1, y_1 = 1 + 0.5*(0+1) = 1.5, x_1 = 0.5, y_2 = 1.5 + 0.5*(0.5 + 1.5) = 2.5, x_2 = 1, y_3 = 2.5 + 0.5*(1+2.5) = 4.25, x_3 = 1.5, y_4 = 4.25 + 0.5*(1.5+4.25) = 7.125, x_4 = 2. Thus, y(2) ≈ 7.125
How do you calculate the change in y at each step?
Multiply the slope at the current point by the step size: Δy = f(x_i, y_i) * h.
What is the final step in approximating y(b) using Euler's method?
After reaching x-value closest to b, the corresponding y-value is the approximate solution.
What is the formula for approximating the change in y () using Euler's Method?
<math-inline>\Delta y = f(x_i, y_i) \cdot \Delta x
What is the formula for finding the next y-value () in Euler's Method?
<math-inline>y_{i+1} = y_i + f(x_i, y_i) \cdot \Delta x
What is the general form of a first-order differential equation?
<math-inline>\frac{dy}{dx} = f(x, y)
What is the formula for absolute error?
<math-inline>| \text{Approximate Value} - \text{Exact Value} |
What is the formula to calculate the next x-value?
<math-inline>x_{i+1} = x_i + h
What is the formula for finding the slope at a given point?
<math-inline>f(x_i, y_i) = \frac{dy}{dx} |_{(x_i, y_i)}
How to calculate the step size, h?
<math-inline>h = \frac{x_{final} - x_{initial}}{n}, where n is the number of steps.
What is the formula for the tangent line approximation?
<math-inline>L(x) = f(a) + f'(a)(x-a)
How do you express the differential equation in terms of y'?
<math-inline>y' = f(x, y)
What is the iterative formula for Euler's method?
<math-inline>y_{n+1} = y_n + h cdot f(x_n, y_n)
What is Euler's Method?
A numerical method to approximate solutions to differential equations with a given initial condition.
What is a differential equation?
An equation that relates a function with its derivatives.
What is an initial condition?
A point (x, y) on the solution curve of a differential equation.
What is step size (h) in Euler's Method?
The constant change in the x-value used in each iteration of Euler's Method.
What does it mean for Euler's method to be a 'first-order' numerical procedure?
It uses the derivative at a single point to approximate the function's value at the next point.
Define numerical solution.
An approximate solution to a differential equation, usually in the form of a table of values.
What is the tangent line approximation?
Using the tangent line at a point to estimate the function's value nearby.
Define absolute error.
The absolute value of the difference between the approximate and exact values.
What is the significance of smaller step sizes in Euler's method?
Smaller step sizes generally lead to more accurate approximations.
What is the purpose of Euler's method?
To approximate the solution of a differential equation when an analytical solution is difficult or impossible to find.