- #1
craighenn
- 5
- 0
Hi, I'd just like to have a quick clarification with regards to the method of undetermined coefficients. I know that if a characteristic equation has the form
(r-4)3 = 0
then the characteristic solution will be
yc = e4t + te4t + t2e4t + t3e4t
and the particular solution ought to be
Y = At4e4t
That all makes sense to me. But I was wondering how the particular solution is to be found if the characteristic solution isn't entirely one root. Like if it was
(r-4)2(r-2) =0
The characteristic solution would be
yc = e4t + te4t + e2t
My question then is two-fold. First, how do I structure the particular solution with regard to the method of undetermined coefficients? Because when all the roots repeat then the particular is just one power of t greater than the greatest one in the solution, like in my first example.
But with two different roots out of 3 (or think of any other example), I'm not sure what to do. Can I even use that method, or is a particular solution even necessary when the characteristic equation has more than just repeated roots?
Hopefully I wrote this clear enough to convey what I'm asking.
(r-4)3 = 0
then the characteristic solution will be
yc = e4t + te4t + t2e4t + t3e4t
and the particular solution ought to be
Y = At4e4t
That all makes sense to me. But I was wondering how the particular solution is to be found if the characteristic solution isn't entirely one root. Like if it was
(r-4)2(r-2) =0
The characteristic solution would be
yc = e4t + te4t + e2t
My question then is two-fold. First, how do I structure the particular solution with regard to the method of undetermined coefficients? Because when all the roots repeat then the particular is just one power of t greater than the greatest one in the solution, like in my first example.
But with two different roots out of 3 (or think of any other example), I'm not sure what to do. Can I even use that method, or is a particular solution even necessary when the characteristic equation has more than just repeated roots?
Hopefully I wrote this clear enough to convey what I'm asking.