3.5 43

[Chad S]

I'm having a difficult time finding the particular solution the book
wants me to find. I've tried quite a few different trig formulas, but
nothing I do seems to lead me in the direction they want. I know my
work is mistake free because wolfram alpha said so. Would anyone mind
leading me down the right path so I can get back to some Differential
Equations?

[Chad S]

Ahh I see now the extra instructions on the homework page. I'll try
not to post after 3 AM to avoid this kind of thing in the future

[Chad S]

well, actually I'm not really sure how to use that hint either.

[Professor Laugesen]

Are you working on part (a) or part (b)?
--
For part (a), you could first try doing a simpler problem of the same
type:

cos(2x) + i sin(2x) = e^(i(2x)) = (e^(ix))^2 = (cos(x) + i
sin(x))^2

Expand the square on the right side. Then equate to what you get to
the left side. You should obtain two "double-angle" formulas, one from
the real part and one from the imaginary part of your equation:
cos(2x) = cos^2(x) - sin^2(x)
sin(2x) = 2sin(x)cos(x)

Moral of the story: you have proved some familiar trig identities in a
new way, by using complex exponentials.

Notice 43(a) is similar, only you now are trying to prove identities
for cos(3x) and sin(3x).

Hint along the way: cos^2(x)=1-sin^2(x)

---

In part (b), you want to use the result of part (a), and then use
Undetermined Coefficients. Note the comment at the bottom of the
Undetermined Coefficients handout about what to do if f(x) on the
right side of the DE consists of a sum of more than one term.