18.1. Fundamental Concepts

Try these first

1. 1.

   Define an Ordinary Differential Equation.

   Hint

   Focus on the function and derivatives.

2. 2.

   What is a numerical method?

   Hint

   Think about how we use methods to calculate values.

3. 3.

   What is the definition of an Ordinary Differential Equation (ODE)?

   • A type of algebraic equation
   • An equation involving functions and their derivatives
   • A polynomial equation

   Hint

   Think about the components of an ODE.

4. 4.

   True or False: Stability ensures that errors grow uncontrollably.

   • True
   • False

   Hint

   Remember the definition of stability.

5. 5.

   Given the ODE y' = -3y with initial condition y(0) = 2, apply Euler’s method with a step size of 0.5 to find the approximation at x = 1.

   Hint

   Pay attention to how you apply the recurrence relation.

6. 6.

   Analyze the convergence of the numerical solution for the ODE y' = -y as h approaches zero, discussing the impact on accuracy.

   Hint

   Consider what happens as you make step sizes smaller.