7.3.2. Step 2: Guess the Form of the Particular Integral

Try these first

1. 1.

   What is the trial solution if f(x) = sin(3x)?

   Hint

   Remember it's sine and cosine.

2. 2.

   If f(x) = 5x^2 + 3x, what's the trial solution?

   Hint

   This is a polynomial function.

3. 3.

   What is the initial guess for the particular integral of e^(3x)?

   • Ae^(3x)
   • Ax*e^(3x)
   • A*cos(3x) + B*sin(3x)

   Hint

   Think of the function form.

4. 4.

   True or False: The trial solution needs to be modified if it overlaps with the complementary function.

   • True
   • False

   Hint

   Consider the implications of having the same terms.

5. 5.

   Solve the differential equation y'' - 3y' + 2y = e^(x) + x. Find the particular integral with appropriate adjustments.

   Hint

   Review the forms of f(x) and identify overlaps.

6. 6.

   Given y'' + y = 4sin(x) + e^(2x), determine the adjustments needed for your trial solutions.

   Hint

   Focus on function types appearing in y_c.