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Here, let's take a look at sums of the fundamental components and/or products of the fundamental components. To do this we'll require the following fact.
Fact- Undetermined Coefficients
If YP1(t) is a particular solution for
y′′ + p (t) y′ + q (t ) y = g1 (t)
And if YP2(t) is an exact solution for
y′′ + p (t ) y′ + q (t) y = g2 (t)
So YP1(t)+ YP2(t) is an exact solution for
y′′ + p (t ) y′ + q (t ) y = g1 (t )+ g2(t)
This fact can be utilized to both find exact solutions to differential equations which have sums in then and to write down guess for functions which have sums in them.
Factoring By Grouping It is a method that isn't utilized all that frequently, but while it can be used it can be somewhat useful. Factoring by grouping can be nice, however it
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