1.7. Methods of Finding Particular Solution

Try these first

1. 1.

   Using the Method of Undetermined Coefficients, what form would you assume for R(x) = 3x^2?

   Hint

   Consider the degree of the polynomial when making your assumption.

2. 2.

   True or False: The Method of Variation of Parameters requires the solution of a homogeneous equation to find u1 and u2.

   Hint

   Recall the relationship between complementary and particular solutions.

3. 3.

   What is the Method of Undetermined Coefficients used for?

   • Finding homogeneous solutions
   • Finding particular solutions
   • Both

   Hint

   Think about what type of solution each method aims to find.

4. 4.

   True or False: The Method of Variation of Parameters can be used when R(x) has any form.

   • True
   • False

   Hint

   Consider the limitations of the Method of Undetermined Coefficients.

5. 5.

   Given the non-homogeneous equation y'' + y = 2e^x, solve for the particular solution using the Method of Undetermined Coefficients.

   Hint

   What happens when you plug your y_p form back into the original equation?

6. 6.

   For the differential equation y'' + 3y' + 2y = cos(x), apply the Method of Variation of Parameters.

   Hint

   Focus on the relationships established from your homogeneous solution!