7.3. Steps in the Method of Undetermined Coefficients

Try these first

1. 1.

   Explain the role of the complementary function in the method of undetermined coefficients.

   Hint

   Consider its definition in relation to the differential equation.

2. 2.

   What is the trial solution if f(x) = sin(ax)?

   Hint

   Think about the forms we use for trigonometric functions.

3. 3.

   What is the first step in the method of undetermined coefficients?

   • Guess the form of y_p
   • Solve the homogeneous equation
   • Substitute into the equation

   Hint

   Think about what comes first in the sequence.

4. 4.

   True or False: The method of undetermined coefficients can be applied to any non-homogeneous function.

   • True
   • False

   Hint

   Remember the conditions listed for the application.

5. 5.

   Solve the equation y'' - 4y' + 4y = 3e^{2x}. Identify your steps.

   Hint

   Be careful with the double root; it changes how you write your complementary function!

6. 6.

   For the equation y'' + 2y' + y = sin(x), find the general solution.

   Hint

   You might need to adjust your trial solution if overlap occurs.