8.1. General Form of a Non-Homogeneous Second-Order Linear Differential Equation

Try these first

1. 1.

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

   Hint

   Think about the structure: it includes derivatives and functions.

2. 2.

   Identify the dependent variable in the equation.

   Hint

   What do we generally measure or calculate in these equations?

3. 3.

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

   • a) y′′ + p(x)y′ + q(x)y = 0
   • b) y′′ + p(x)y′ + q(x)y = g(x)
   • c) y′ + p(y)y + q(x) = g(x)

   Hint

   Look for the term that represents external input.

4. 4.

   True or False: The function p(x) represents a non-homogeneous term.

   • True
   • False

   Hint

   Think about the roles of each variable in the equation.

5. 5.

   Solve the equation y′′ + 3y′ + 2y = sin(x) and explain each part of your solution.

   Hint

   Identify each function and consider the application of particular solutions.

6. 6.

   Describe how changing g(x) to a polynomial affects the overall solution.

   Hint

   Reflection on the form of g(x) leads to varied methods of finding fields of applications.