Hi everyone,

As you work through WeBWorK #9, you may run into a few instances where it is not obvious what guess to make for the particular solution to the nonhomogeneous equation.  Here are two tips that might help:

 

What if the right side has both an exponential and a trig function?

If the right side of your differential equation has form similar to:  e^{ax} \cos(bx)

then your guess should have the form:  Ae^{ax} \sin(bx) + Be^{ax} \cos(bx)

 

What if the right side is a solution to the complementary equation (where the complementary equation has a repeated root)?

If your initial guess:  Ae^{ax} is a solution to the complementary equation, we adjust it by multiplying by x: Axe^{ax}.

If Axe^{ax} is *also* a solution to the complementary equation (due to a repeated root), adjust it by multiplying by x again:  Ax^2e^{ax}.

 

Let me know if you run into other weird things.  Questions are natural, and welcome!

Prof. Reitz