All Flashcards
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 are the differences between Euler's Method and the exact solution of a differential equation?
Euler's Method: Provides an approximate numerical solution. Exact Solution: Provides the precise analytical solution.
What are the differences between using a smaller versus a larger step size in Euler's Method?
Smaller step size: More accurate, more computation. Larger step size: Less accurate, less computation.
What are the differences between Euler's Method and Improved Euler's Method (Heun's Method)?
Euler's Method: Uses the slope at the beginning of the interval. Improved Euler's Method: Averages slopes at the beginning and end of the interval for better accuracy.
What are the differences between Euler's Method and Runge-Kutta methods?
Euler's Method: First-order, less accurate. Runge-Kutta: Higher-order, more accurate.
What are the differences between numerical and analytical solutions?
Numerical: Approximate, discrete values. Analytical: Exact, continuous function.
What are the differences between local and global error in numerical methods?
Local: Error in a single step. Global: Accumulated error over multiple steps.
What are the differences between explicit and implicit numerical methods?
Explicit: Uses known values to calculate the next value. Implicit: Uses unknown values, requiring solving an equation.
What are the differences between forward and backward Euler methods?
Forward: Uses the slope at the beginning of the interval. Backward: Uses the slope at the end of the interval.
What are the differences between Euler's method and Taylor series method?
Euler's Method: Uses only the first derivative term. Taylor Series: Uses multiple derivative terms for higher accuracy.
What are the differences between Euler's method and the midpoint method?
Euler's Method: Approximates using the slope at the beginning of the interval. Midpoint Method: Approximates using the slope at the midpoint of the interval.
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.