2.7. Examples

Try these first

1. 1.

   Solve the equation d²y/dx² - 2dy/dx + y = 0.

   Hint

   Look for repeated roots in the auxiliary equation.

2. 2.

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

   Hint

   Check for complex roots in the auxiliary equation.

3. 3.

   What is the general solution of d²y/dx² - 5dy/dx + 6y = 0?

   Hint

   Derive it from the auxiliary equation.

4. 4.

   True or False: The general solution for repeated roots includes a linear term.

   • True
   • False

   Hint

   Recall the formula for repeated roots.

5. 5.

   Solve the differential equation d²y/dx² - 8dy/dx + 16y = 0 and analyze the implications of the solution.

   Hint

   Pay attention to the nature of the roots and apply the correct form.

6. 6.

   Evaluate the equation d²y/dx² + 6dy/dx + 10y = 0, identify the nature of the roots, and provide the general solution.

   Hint

   Utilize the quadratic formula and consider the implications of complex roots.