7.7. Special Cases and Modifications

Try these first

1. 1.

   What adjustment is needed when the trial solution overlaps with the complementary function?

   Hint

   Consider how to eliminate duplication.

2. 2.

   What is a non-standard right-hand side?

   Hint

   Think of unusual functions.

3. 3.

   When faced with repeated roots, how can you modify your guess?

   • Multiply by x
   • Add a constant
   • Differential term

   Hint

   Think about how duplicates are formed.

4. 4.

   True or False: The undetermined coefficients method can be used for rational functions like 1/(x+1).

   • True
   • False

   Hint

   Consider which functions fit standard forms.

5. 5.

   Solve the differential equation y′′ + 4y = cos(2x), ensuring to adjust your trial character.

   Hint

   Identify the nature of your complement.

6. 6.

   For y′′ − 3y′ + 2y = x^3 sin(x), describe how to approach this using variation instead of undetermined coefficients.

   Hint

   Reflect on core concepts of replacement methods.