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Fourier series and differential equations
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[QUOTE="vela, post: 5651760, member: 221963"] You need to be more careful. You have ##y(t) = \sum_n c_n e^{int}##. When you plug this into the differential equation, you end up with $$\sum_{n=-\infty}^\infty -n^2c_n e^{int} + \sum_{n=-\infty}^\infty ac_n e^{int} = \sum_{n=-\infty}^\infty (-1)^n c_n e^{int}.$$ [SIZE=4]The orthogonality of the Fourier components then implies that[/SIZE] $$(-n^2+a) c_n = (-1)^n c_n$$ [SIZE=4]for all ##n##. There are two ways this relationship can hold: ##a=n^2 + (-1)^n## or ##c_n = 0##.[/SIZE] [SIZE=4]Now remember that you're looking at the situation where ##a## is a constant, so it can't vary with ##n##. Suppose ##a=3##. You can see there is no integer value of ##n## such that ##3 = n^2 + (-1)^n##, so the only way the relationship holds is ##c_n = 0## for all ##n##. In other words, ##y=0##, the trivial solution.[/SIZE] But suppose ##a=5 = 2^2 + (-1)^2##. The solution has to satisfy $$\sum_{n=-\infty}^\infty -n^2c_n e^{int} + \sum_{n=-\infty}^\infty [2^2+(-1)^2]c_n e^{int} = \sum_{n=-\infty}^\infty (-1)^n c_n e^{int},$$ which implies $$(-n^2+5) c_n = (-1)^n c_n$$ for all ##n##. What happens when ##n=\pm2##? What about when ##n \ne \pm 2##? What does the series look like in this case? [/QUOTE]
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Fourier series and differential equations
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