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Friday, January 1, 2016

The method of undetermined coefficients (I)

Problem:

$ y'' - y = \cos x $

Solution:

Let's first solve $ z'' - z  = 0 $, using the characteristic polynomial method, we know the answer is $ z = Ae^x + Be^{-x} $.

It remains to find just one function that fits $ w'' - w = \cos x $, then we know $ y = w + z $ and have the degree of freedom we need.

At this point, the so called "method of undetermined coefficients" is really just about guessing the solution, let's guess our solution is $ w = \cos x $ and see

$ (\cos x)'' - \cos x = -2 \cos x $.

So now the solution is obviously $ w = \frac{-1}{2}\cos x $

The full solution is then $ Ae^x + Be^{-x} - \frac{1}{2}\cos x $.

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