WeBWorK 9: Tips for guessing solutions to nonhomogeneous equations

April 1, 2019 / Jonas Reitz / 0 Comments

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

Resources

homework hints nonhomogenous

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WeBWorK 9: Tips for guessing solutions to nonhomogeneous equations

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

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

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