7.4. Illustrative Examples

Try these first

1. 1.

   What is the complementary function for y′′ + 4y = 0?

   Hint

   Use the auxiliary equation approach.

2. 2.

   What is a typical guess for the particular integral if f(x) = e^(3x)?

   Hint

   Think about the form of the non-homogeneous term.

3. 3.

   What does the method of undetermined coefficients help us find?

   • Complementary Function
   • Particular Integral
   • General Solution

   Hint

   Remember the names of different components in a solution.

4. 4.

   True or False: The trial solution is always unique in the method of undetermined coefficients.

   • True
   • False

   Hint

   Think about how we adjust our trial solutions.

5. 5.

   Given the differential equation y′′ + 5y′ + 6y = e^(-3x), find the general solution.

   Hint

   Start by solving the homogeneous part, then adjust your guess for y_p.

6. 6.

   How would you solve the equation y′′ + 4y = x^2 cos(2x) using the method of undetermined coefficients?

   Hint

   Remember to check for duplication with the complementary function.