18.1.2. Numerical Method

Try these first

1. 1.

   Define a numerical method.

   Hint

   Think about how we can calculate using previous values.

2. 2.

   What does the step size $h$ indicate?

   Hint

   A smaller step size generally leads to more accuracy.

3. 3.

   What is a numerical method?

   • A way of exact solutions
   • An approximation procedure
   • Only for linear equations

   Hint

   Think about how we address complex ODEs.

4. 4.

   True or False: A smaller step size results in larger local truncation error.

   • True
   • False

   Hint

   Recall the relationship between step size and accuracy.

5. 5.

   Formulate a numerical method to solve the ODE $y' = -3y$ with the initial condition $y(0) = 2$ using h = 0.5.

   Hint

   Pay attention to the signs while calculating the next values.

6. 6.

   Analyze the stability of the numerical method applied to the equation $y' = -y$. What does your analysis suggest about step size?

   Hint

   Consider the implications for large values of h.