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It is the full blown case where we consider every final possible force which can act on the system. The differential equation in this case,
Mu'' + γu' + ku = F( t)
The displacement function here will be
u(t) = uc(t) + UP(t)
Here the complementary solution will be the solution to the free, damped case and the exact solution will be found using undetermined coefficients or variation of parameter that ever is most convenient to utilize.
There are a couple of things to see now about this case. First, from our work back into the free, damped case we identify that the complementary solution will come to zero as t increases.
Due to this the complementary solution is often termed as the transient solution in this case. Also, due to this behavior the displacement will start to look more and more like the exact solution as t raises and so the particular solution is frequently termed as the steady state solution or forced response.
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It is not the first time that we've looked this topic. We also considered linear independence and linear dependence back while we were looking at second order differential equation
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