8.2. Principle of the Variation of Parameters

Try these first

1. 1.

   Define a non-homogeneous equation.

   Hint

   Think about what extra terms might look like.

2. 2.

   What is the form of a homogeneous solution?

   Hint

   Consider the constant coefficients for a family of solutions.

3. 3.

   What is the first step in the Variation of Parameters?

   • Identify g(x)
   • Differentiate y_h
   • Assume a form for y_p

   Hint

   Think about how we structure our assumptions.

4. 4.

   True or False: The Variation of Parameters can only handle polynomial forcing functions.

   • True
   • False

   Hint

   Consider the flexibility of the method.

5. 5.

   Solve the following differential equation using the variation of parameters: y'' + 2y' + y = e^x.

   Hint

   Remember to check your integration carefully for any retries.

6. 6.

   Using the Variation of Parameters, show how to derive a particular solution for y'' - 4y = sin(x).

   Hint

   Start fresh with the Wronskian calculation and methodically go through each step.