How do you use Inverse Operators for n'th order DEs?

May 2010
3
0
Actually, I didn't really mean n't order as in generalized forms or anything, just higher order than 1, but not so high as to come computationally ridiculous or require any upper division math major stuff. The final's coming up and I kind of totally forget how to use inverse operators to non-homogeneous solve n'th order differential equations. My teacher taught this method instead of undetermined coefficients to solve certain DE's, I can't find the handout he gave us it isn't in the book or anywhere else on the internet as far as I can find (I'm sure it's there somewhere, I just can't find it). Any information on this subject would be appreciated.

Also, although I don't remember it, this method seemed to be a relatively straightforward method of solving these DEs and my teacher found preferable to undetermined coefficients. It might be a good idea for someone to add this to the Differential Equations Tutorial thread. (Maybe I'll do it after my final if I ever find that handout (Worried) )
 
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Chris L T521

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Actually, I didn't really mean n't order as in generalized forms or anything, just higher order than 1, but not so high as to come computationally ridiculous or require any upper division math major stuff. The final's coming up and I kind of totally forget how to use inverse operators to non-homogeneous solve n'th order differential equations. My teacher taught this method instead of undetermined coefficients to solve certain DE's, I can't find the handout he gave us it isn't in the book or anywhere else on the internet as far as I can find (I'm sure it's there somewhere, I just can't find it). Any information on this subject would be appreciated.

Also, although I don't remember it, this method seemed to be a relatively straightforward method of solving these DEs and my teacher found preferable to undetermined coefficients. It might be a good idea for someone to add this to the Differential Equations Tutorial thread. (Maybe I'll do it after my final if I ever find that handout (Worried) )
The only other method that I know besides undetermined coefficients is the annihilator method as seen in my tutorial here.

I prefer this technique over undetermined coefficients because I can figure out the exact form of our particular solution (instead of guessing it) when solving the non-homogeneous ODE. Take note, though, that it only works on certain \(\displaystyle f(x)\) in the DE \(\displaystyle g(y,y^{\prime},\ldots,y^{(n)})=f(x)\). For the case of other more complex \(\displaystyle f(x)\) terms, we apply variation of parameters (There may be some generalized form for higher order equations, but the ideas should be the same).

Does this help?