1.6.2. Complete Solution

Try these first

1. 1.

   What does the Complete Solution of a linear differential equation consist of?

   Hint

   Think about both parts we discussed in class.

2. 2.

   Define the term Complementary Function.

   Hint

   This does not include any external forces.

3. 3.

   What are the two main components of the Complete Solution?

   • y_c and y_p
   • y_c only
   • y_p only

   Hint

   Consider what each part represents.

4. 4.

   True or False: The Complementary Function accounts for external forces acting on the system.

   • True
   • False

   Hint

   What does 'Complementary' indicate in its name?

5. 5.

   Given the equation d²y/dx² + 4dy/dx + 4y = sin(x), determine the Complete Solution. Start with finding the Auxiliary Equation.

   Hint

   The repeated root indicates a specific form for y_c.

6. 6.

   Solve the non-homogeneous linear differential equation d²y/dx² - 3dy/dx + 2y = e^x. Find both parts of the Complete Solution.

   Hint

   The form of y_p reflects the nature of the non-homogeneous term.