6.9. Conceptual Notes

Try these first

1. 1.

   What is the primary difference between homogeneous and non-homogeneous differential equations?

   Hint

   Think about the presence or absence of external forces.

2. 2.

   Define the term 'forcing function' in the context of differential equations.

   Hint

   It's what pushes the system away from its natural response.

3. 3.

   What do non-homogeneous differential equations primarily model?

   • Natural vibrations
   • Forced systems
   • Linear motion

   Hint

   Think about how forces change the system's behavior.

4. 4.

   True or False: The method of undetermined coefficients can be applied to all types of forcing functions.

   • True
   • False

   Hint

   Remember the limits of P.E.T.

5. 5.

   Solve the non-homogeneous differential equation: y'' + 3y' + 2y = e^x.

   Hint

   Start with characteristics before tackling the PI.

6. 6.

   Using the method of variation of parameters, solve: y'' + y = sin(2x).

   Hint

   Split the problem into manageable parts!