7.5 Approximating Solutions Using Euler’s Method

• Euler's method uses the slope of the tangent line to estimate the next value of the solution


  True
• What is the formula used in Euler's Method to approximate the solution to an initial value problem?

  y_{n + 1} = y_{n} + f(x_{n}, y_{n}) \Delta x</latex>
• The exact solution to a differential equation is given by the formula $y =$$f(x)$
• The accuracy of Euler's Method improves as the step size $\Delta x$ decreases
• What does Euler's Method use to estimate the next value of the solution at each step?

  The slope of the tangent line
• What does $y_{n}$ represent in Euler's Method?


  Current value of solution
• What does $\Delta x$ signify in Euler's Method?


  Step size
• What is the formula used in Euler's Method to approximate the next value of the solution?

  y_{n + 1} = y_{n} + f(x_{n}, y_{n}) \Delta x</latex>
• What is an initial value problem in the context of Euler's Method?

  Differential equation with initial condition
• Euler's Method provides an approximate solution, while the exact solution formula provides the true, analytical solution.
• In the example, what is the initial value problem given?

  $\frac{dy}{dx} =$$y - x$, $y(0) =$$2$
• What is the value of $y_{1}$ after the first iteration of Euler's Method in the example?


  $y_{1} =$$3$
• What is the main purpose of Euler's Method?

  Approximate solutions
• How does the accuracy of Euler's Method change as the step size decreases?

  Improves
• The formula used in Euler's Method to approximate the solution of an initial value problem is: $y_{n + 1} =$$y_{n} +$$f(x_{n}, y_{n}) \Delta x$
• Euler's Method calculates the exact solution of an initial value problem

  False
• Steps to apply Euler's Method

  1️⃣ Define the initial value problem

  2️⃣ Choose a step size $\Delta x$

  3️⃣ Use the formula $y_{n + 1} =$$y_{n} +$$f(x_{n}, y_{n}) \Delta x$

  4️⃣ Approximate the next solution value
• In Euler's Method, the step size $h$ is the increment by which we increase the independent variable x
• What happens to computational effort as the step size decreases in Euler's Method?

  It increases
• A larger step size in Euler's Method requires less computational effort but reduces accuracy


  True
• Runge-Kutta methods are more accurate than Euler's because they use multiple slope evaluations within each step
• For what type of problems is Euler's Method most suitable?

  Quick estimations
• Euler's Method is a numerical technique for approximating the solution to an initial value problem
• The accuracy of Euler's method improves as the step size is decreased
• Euler's Method calculates the exact solution to a differential equation

  False
• What does Euler's Method approximate using the slope of the tangent line?

  The next value of the solution
• Euler's Method provides an approximate solution, while the exact solution formula provides the true solution


  True
• Euler's Method depends on the step size to improve accuracy
• In Euler's Method, $f(x_{n}, y_{n})$ represents the slope of the tangent line at the current point
• Euler's Method calculates the exact solution of a differential equation.

  False
• Euler's Method approximates the solution using the slope
• The accuracy of Euler's Method improves as the step size increases.

  False
• What is the key idea behind Euler's Method for approximating solutions?

  Use tangent line slopes
• To approximate $y(1)$ using Euler's Method with \Delta x = 0.5</latex>, the first step is to define the initial conditions
• The approximate solution y(1) \approx 4.25</latex> is obtained using Euler's Method with $\Delta x =$$0.5$.


  True
• An initial value problem consists of a differential equation and an initial condition
• Match the characteristic with the method:

  Euler's Method ↔️ Approximates using tangent lines

  Exact Solution ↔️ Calculates precise solution
• What does $y_{n + 1}$ represent in Euler's Method?


  Approximation at $x_{n + 1}$
• The accuracy of Euler's Method improves with a smaller step size
• What is the initial value in the example problem \frac{dy}{dx} = y - x</latex> with $y(0) =$$2$?


  2