7.2.2. Euler’s Method

Try these first

1. 1.

   Explain Euler's Method in your own words.

   Hint

   Think about how you would describe the basic steps.

2. 2.

   What is the importance of the step size in Euler's Method?

   Hint

   Consider what happens when you take bigger or smaller steps.

3. 3.

   What is Euler's Method primarily used for?

   • Finding analytical solutions
   • Approximating solutions to ODEs
   • Deriving derivatives

   Hint

   Think about the scenarios where analytical solutions are hard to find.

4. 4.

   True or False: The smaller the step size in Euler's Method, the more accurate the approximation.

   • True
   • False

   Hint

   Consider how precision typically relates to the size of steps.

5. 5.

   You are given the ODE $\frac{dy}{dx} = 3y$ with $y(0) = 1$. Using Euler's Method with a step size of $h = 0.05$, approximate the value of $y(0.2)$. Show your work.

   Hint

   Keep track of each value and how $f$ changes at every step.

6. 6.

   Apply Euler's Method to $\frac{dy}{dx} = x^2 - y$ with $y(0) = 0$. Find $y(0.1)$ and $y(0.2)$ using $h = 0.1$. Describe any changes you notice in your process.

   Hint

   Pay attention to how the function's form affects the slope and resulting $y$ values.