3.10. Problems for Practice

Try these first

1. 1.

   Solve the equation d²y/dx² + 3dy/dx + 2y = 0 where y(0) = 1 and y'(0) = 0.

   Hint

   Start by forming the characteristic equation.

2. 2.

   Classify the roots for y'' + y' + y = 0.

   Hint

   Calculate the discriminant.

3. 3.

   What is the characteristic equation for d²y/dx² + 3dy/dx + 2y = 0?

   • r² + 3r + 2 = 0
   • r² - 3r + 2 = 0
   • r² + 2r + 3 = 0

   Hint

   Identify coefficients from the standard form.

4. 4.

   True or False: The discriminant can determine the type of roots.

   • True
   • False

   Hint

   Review how discriminants classify roots.

5. 5.

   Prove how changing damping ratios in a cantilever beam affect the solutions of its oscillatory system.

   Hint

   Start with the governing equations for both underdamped and overdamped systems.

6. 6.

   Explain how the method of undetermined coefficients could be applied to a second-order non-homogeneous equation and solve it.

   Hint

   Identify the non-homogeneous parts to apply the method accurately.