6.1. General Form of a Linear Non-Homogeneous Differential Equation

Try these first

1. 1.

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

   Hint

   Look for the terms that indicate the order and type.

2. 2.

   Define a complementary function.

   Hint

   Think about the solution when the right side of the equation is zero.

3. 3.

   What is a key distinguishing feature of a non-homogeneous differential equation?

   • It is always linear.
   • It includes a forcing function.
   • It has constant coefficients.

   Hint

   Consider what distinguishes these types of equations.

4. 4.

   True or False: The complementary function describes the natural response of the system.

   • True
   • False

   Hint

   Think about the context of a homogeneous equation.

5. 5.

   Derive both the complementary function and particular integral for the equation $y'' + 4y' + 4y = e^{-2x}$, and explain the significance of each part.

   Hint

   Start with the characteristic equation for the CF and derive the correct guess for the PI.

6. 6.

   Using the method of variation of parameters, solve the equation $y'' + y = cos(t)$.

   Hint

   Ensure you use the correct form and accounts for the coefficients in your calculations.