10.. Numerical Solutions of ODEs

Try these first

1. 1.

   What is the initial step in the Modified Euler’s Method?

   Hint

   Think about how slopes represent the rate of change.

2. 2.

   True or False: The Modified Euler’s Method is a first-order numerical technique.

   Hint

   Recall the accuracy level compared to basic Euler’s Method.

3. 3.

   What is the primary advantage of the Modified Euler’s Method?

   • It is the most accurate method.
   • It requires fewer computations.
   • It improves upon basic Euler’s method.

   Hint

   Think about why basing on averages instead of single slopes helps.

4. 4.

   True or False: The Modified Euler's method has only one slope calculation per step.

   • True
   • False

   Hint

   Recall the steps involved in predicting and correcting.

5. 5.

   Analytically find the solution to the equation dy/dx = 2 - y, with an initial condition y(0)=1 and subsequently use Modified Euler’s Method with h=0.1 to approximate y(0.5). Compare the values and explain the differences.

   Hint

   Focus on both solving the analytical equation and implementing numerical steps iteratively.

6. 6.

   Explore the implications of step size on the accuracy of Modified Euler's Method. What would happen if step size h were halved? Perform an analysis between h=0.1 and h=0.05 using the same initial value problem.

   Hint

   Compare your results closely from both methods while adjusting h.