(a) Write
cos 3x + i sin 3x = e3ix = (cos x + i sin x)3
by Euler’s formula, expand, and equate real and imaginary parts to derive the identities
cos3 x = ¾ cos x + ¼ cos 3x,
sin3 x = ¾ sin x − ¼ sin 3x.
(b) Use the result of part (a) to find a general solution of
y″ + 4y = cos3 x.
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