Nonhomogeneous second order linear differential equations

[Ortix]

Homework Statement

i'm supposed to find the general solution of the equation: y'' + 3y' + 2y = e^x + e^-x

Homework Equations

I have no problem with solving this equation however, i am confused with the step they are taking in the solutions (circled):
http://img522.imageshack.us/img522/7122/calcwtf.jpg

Where does that x come from next to the B? I have been looking in my book, but there is no example with this kind of solution (right hand side)

The Attempt at a Solution

i got up to the auxiliary equation and i get stuck with finding the particular solution. I'm not sure what to do here... For single variable and products as the solution i have no problem.. the addition is no biggie either (i just separate them) but the step shown above just throws me off.

help is obviously very much appreciated! :)

 

[tiny-tim]

Hi Ortix!

(try using the X² icon just above the Reply box )

y'' + 3y' + 2y = e^x + e^-x
…
Where does that x come from next to the B?

As you obviously know, when e^kx is on the RHS, you generally try a multiple of it as part of a particular solution.

however, in this case, the fact that e^kx (with k = -1) is already a solution to the LHS means that it can't work …

in this case, we "go up one" and use xe^kx instead

(and if k was an nth root for the LHS, we'd use xⁿe^kx )

 

[gomunkul51]

It is a part of a system called "Method of undetermined coefficients" for finding a particular solution for a constant coefficient linear ODE.

Read it thorough and you will know that the template is the correct template for finding a particular solution to that ODE.

 

[Ortix]

i'm still not TOO clear about this.. i will follow up on gomunkul51's suggestion and read about that. I found it in my book. If i experience more difficulties, I shall return to the land of the smart =).. next year tho!

Happy new year everyone, I'm out to celebrate!