7.1. Overview of Linear Non-Homogeneous Differential Equations

Try these first

1. 1.

   Define a linear non-homogeneous differential equation.

   Hint

   Think about the general form with a forcing function.

2. 2.

   What are the two parts of the general solution?

   Hint

   Consider what each part represents.

3. 3.

   What is a linear non-homogeneous differential equation?

   • An equation with no forcing function
   • An equation with a non-zero term
   • An equation with constant coefficients only

   Hint

   Look for the term that differentiates it from homogeneous equations.

4. 4.

   True or False: The particular solution is only relevant for specific standard forms of forcing functions.

   • True
   • False

   Hint

   Remember the types of functions we can use.

5. 5.

   Given the differential equation $y'' + 4y = e^{-x}$, determine the complementary function and apply the method of undetermined coefficients to find the particular solution.

   Hint

   Don't forget to check the roots of the auxiliary equation!

6. 6.

   For $y'' + 3y' + 2y = x^2 + e^x$, solve for the complementary function and particular solutions for both terms.

   Hint

   Remember the specific forms for your guesses!