3. Second-Order Homogeneous Equations with Constant Coefficients

Try these first

1. 1.

   What is the general form of a second-order homogeneous linear differential equation?

   Hint

   Look for the terms and their coefficients.

2. 2.

   Define the term 'homogeneous'.

   Hint

   Think about the implications of zero.

3. 3.

   What is the characteristic equation derived from the second-order linear differential equation?

   • ar^2 + br + c = 0
   • d^2y/dx^2 = 0
   • y = C e^{kx}

   Hint

   Think about substitution.

4. 4.

   True or False: The general solution of a second-order homogeneous equation can be formed with exponential terms only.

   • True
   • False

   Hint

   Recall the nature of the roots.

5. 5.

   Derive the characteristic equation for the second-order linear ODE $y'' - 7y' + 10y = 0$, classify the roots, and find the general solution.

   Hint

   Start by substituting your assumed solution.

6. 6.

   Model the free vibration of a cantilever beam described by the ODE $m \frac{d^2y}{dt^2} + c \frac{dy}{dt} + k y = 0$. Derive and solve for the response.

   Hint

   Identify damping cases based on the roots.