4.2. Case of Complex Roots (When D < 0)

Try these first

1. 1.

   What is the form of the general solution when the roots are complex?

   Hint

   Think about how we express solutions in terms of cosine and sine.

2. 2.

   What indicates damped oscillation in a differential equation?

   Hint

   Remember, it has to do with the value of α.

3. 3.

   What does a negative discriminant indicate in the characteristic equation?

   • Real roots
   • Complex roots
   • No roots

   Hint

   Recall the formula for the discriminant.

4. 4.

   True or False: If α < 0, then the amplitude of oscillation remains constant.

   • True
   • False

   Hint

   Think about the nature of damped oscillations.

5. 5.

   Given the differential equation d²y/dx² + 5dy/dx + 6y = 0, derive the general solution and discuss its implications for dynamic structures.

   Hint

   Calculate the discriminant to confirm root types.

6. 6.

   A high-rise building sways under wind with a natural frequency ω = 2π rad/s and damping ratio ζ = 0.2. Calculate the damped frequency and interpret the results.

   Hint

   Use the formula for damped frequency and consider how damping affects real-world applications.