6. Non-Homogeneous Equations

Try these first

1. 1.

   Write the general form of a second-order linear non-homogeneous differential equation.

   Hint

   Remember to include the forcing function on the right side.

2. 2.

   What is the complementary function?

   Hint

   Think about the behavior when no external forces act.

3. 3.

   What is the primary characteristic of a non-homogeneous differential equation?

   • It has no forcing function.
   • It includes external influences.
   • It only describes natural behavior.

   Hint

   Think about what distinguishes it from a homogeneous equation.

4. 4.

   True or False: The method of undetermined coefficients can only be used if f(x) is polynomial.

   • True
   • False

   Hint

   Consider all forms of functions that fit in the method.

5. 5.

   A beam is subjected to a periodic load described by f(x) = f_0 sin(ωt). Derive the appropriate non-homogeneous differential equation and discuss the implications for resonance.

   Hint

   Start by presenting the loading conditions and recall methods related to harmonic functions.

6. 6.

   Given the equation d²y/dx² + 2 dy/dx + 3y = e^(-x) + sin(x), solve it by applying both methods discussed: undetermined coefficients and variation of parameters, and compare results.

   Hint

   Think systematically about how you structure each method towards finding the particular integral.