3.2. Characteristic Equation

Try these first

1. 1.

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

   Hint

   Only consider the coefficients of the original equation.

2. 2.

   What happens when the discriminant is negative?

   Hint

   Think about how the roots can affect the solution's behavior.

3. 3.

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

   • r² + 2r + 1 = 0
   • r² - 2r + 1 = 0
   • r² + 2r - 1 = 0

   Hint

   Look carefully at how the differential equation coefficients correspond to terms in the characteristic equation.

4. 4.

   If the discriminant is zero, what type of roots do we have?

   • True
   • False

   Hint

   Think about what this implies for the general solution.

5. 5.

   Derive the general solution for the ODE d²y/dx² - 6dy/dx + 9y = 0.

   Hint

   Use the quadratic formula and remember the implications of multiple roots.

6. 6.

   For the differential equation y'' + 4y' + 20y = 0, find and analyze the nature of its roots and the general solution.

   Hint

   Identify the real and imaginary parts carefully to construct the final solution.