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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.
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