2.2. Homogeneous Linear Equations with Constant Coefficients

Try these first

1. 1.

   Solve the equation: $\frac{d^2y}{dx^2} + 7\frac{dy}{dx} + 12y=0$.

   Hint

   Start with the auxiliary equation.

2. 2.

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

   Hint

   Recall the definition of the equation.

3. 3.

   What characterizes a homogeneous linear equation?

   • It has constant coefficients.
   • It equals a non-zero constant.
   • All terms are linear.
   • Both 1 and 3.

   Hint

   Reflect on the definition.

4. 4.

   True or False: The general solution of a second-order linear homogeneous ODE contains three arbitrary constants.

   • True
   • False

   Hint

   Recall the number of roots affecting the solution.

5. 5.

   Solve the differential equation: $\frac{d^2y}{dx^2} + 5\frac{dy}{dx} + 6y = 0$. Determine the nature of the roots and interpret the solution.

   Hint

   Look for factors of the characteristic equation.

6. 6.

   Consider the system described by $\frac{d^2y}{dx^2} + 2y = 0$. Find its general solution and discuss what this means in terms of oscillation.

   Hint

   Recall how complex numbers relate to oscillatory behavior.