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(adsbygoogle = window.adsbygoogle || []).push({});

(r-4)^{3}= 0

then the characteristic solution will be

y_{c}= e^{4t}+ te^{4t}+ t^{2}e^{4t}+ t^{3}e^{4t}

and the particular solution ought to be

Y = At^{4}e^{4t}

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

y_{c}= e^{4t}+ te^{4t}+ e^{2t}

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.

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# Method of Undetermined Coefficients for Higher Order Linear Equations

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