4.5. Example Problems

Try these first

1. 1.

   Solve the equation d²y + 6dy + 10y = 0.

   Hint

   Use the quadratic formula to find the roots.

2. 2.

   What does a negative discriminant indicate in a differential equation?

   Hint

   Recall the discriminant formula D = b² - 4ac.

3. 3.

   What is the characteristic equation for the differential equation d²y + 4dy + 13y = 0?

   • r² + 4r + 13 = 0
   • r² - 4r + 13 = 0
   • r² + 4r - 13 = 0

   Hint

   Recall the format of the general second-order linear differential equations.

4. 4.

   True or False: A negative discriminant indicates real and distinct roots.

   • True
   • False

   Hint

   Think about the relationship between the discriminant and the nature of the roots.

5. 5.

   A suspension bridge experiences oscillations when wind blows towards it. If the governing equation is d²y/dt² + 8dy/dt + 16y = 0, analyze the system by finding the roots and assessing stability.

   Hint

   Use the characteristic equation to derive the roots and analyze their real part.

6. 6.

   For a damped harmonic oscillator with a mass of 3 kg, damping coefficient of 6 Ns/m, and spring constant of 75 N/m, calculate the damping ratio ζ and natural frequency ω_n.

   Hint

   Use the formulas for ν and rooftop damping calculations.