10.1.6. Worked-Out Example

Try these first

1. 1.

   What value of $k_1$ do we compute in the first iteration if $x_0 = 0$ and $y_0 = 1$?

   Hint

   Evaluate the function at the initial point.

2. 2.

   If the initial step size is $h = 0.1$, what is the predicted value $y^*$ after the first iteration?

   Hint

   Use the formula for prediction with the calculated slope.

3. 3.

   What is the main advantage of the Modified Euler's Method over the standard Euler's Method?

   • It is easier to compute
   • It uses only one slope
   • It improves accuracy by averaging slopes

   Hint

   Think about how multiple data points affect estimates.

4. 4.

   True or False: The Modified Euler's Method requires two function evaluations for each step.

   • True
   • False

   Hint

   Recall how we calculated both slopes.

5. 5.

   Given the differential equation $\frac{dy}{dx} = y - x^2 + 1$ with an initial condition $y(0) = 0.5$ and step size $h = 0.2$, calculate $y(0.2)$ using the Modified Euler’s Method.

   Hint

   Remember to calculate both slopes at each step just like in our example.

6. 6.

   Discuss how the error in the Modified Euler's Method can accumulate across iterations. Illustrate it with an example of taking too large of a step size.

   Hint

   Think about how estimates differ when you leap over more points on the function.