2.10. Solved Exercises

Try these first

1. 1.

   Solve the equation d²y/dx² + 4y = 0.

   Hint

   Look for the characteristic roots involved.

2. 2.

   What form of solution corresponds to repeated roots?

   Hint

   Consider the multiplicity of the roots.

3. 3.

   What does the auxiliary equation allow us to find?

   • Roots of the equation
   • Boundary conditions
   • Specific functions

   Hint

   It relates to solving the differential equation.

4. 4.

   True or False: Complex roots yield exponential solutions.

   • True
   • False

   Hint

   Consider the form of the solutions for oscillatory systems.

5. 5.

   Solve the differential equation d²y/dx² - 6dy/dx + 9y = 0 and interpret the solution.

   Hint

   Identify roots of the auxiliary equation first.

6. 6.

   If we have a structure modeled by the equation d²y/dx² + 8dy/dx + 16y = 0, find the general solution.

   Hint

   Look at the discriminant for root types.