## Differential equation – Part III Fourier series

Fourier Series To solve the ODE ${y^{\prime\prime}} + a{y^{\prime}} + by = f\left( t \right)$, in which $f\left( t \right)$ is often the combination of ${e^t}$, $\sin t$ and $\cos t$, then any reasonable $f\left( t \right)$ which is periodic, can be discontinuous but not terribly discontinuous, can be written as Fourier Series. $f\left( t \right) = {c_0} + \sum\limits_{n = 1}^\infty {{a_n}\cos nt + {b_n}\sin nt}$ Theorem: $u\left( t \right),v\left( t \right)$ are functions with period $2\pi$ on the real number,

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