2.9. Exercises

Try these first

1. 1.

   Solve the equation d²y/dx² + 5y = 0. What type of roots do you expect?

   Hint

   Use the auxiliary equation.

2. 2.

   What is the general solution when you have real distinct roots?

   Hint

   Recall the roots format.

3. 3.

   What form does a second-order linear homogeneous ODE take?

   • dy/dx + py + q = 0
   • d²y/dx² + p dy/dx + qy = 0
   • d²y/dx² = 0

   Hint

   Identify the equality to zero.

4. 4.

   Does the general solution of a second-order ODE have arbitrary constants?

   • True
   • False

   Hint

   Think about the roots of the auxiliary equation.

5. 5.

   Given the equation d²y/dx² + 10 = 0, derive the solution and determine the type of roots.

   Hint

   Convert the equation into auxiliary form.

6. 6.

   For d²y/dx² + 8dy/dx + 16y = 0, prove that the roots are repeated and write the general solution.

   Hint

   Look at the discriminant to confirm root multiplicity.