18.034, Honors Differential Equations
Prof. Jason Starr
Lecture 17
3/12/04
1. Philosophy about what Fourier series are supposed to do and why this is useful for us:
y” + ay + by = f(t)
If f(t) is periodic of period 2L, write
∞
f(t) =
A0 +
∑A
n
n =1
⎛ nπt ⎞
⎛ nπt ⎞
cos⎜
⎟ +
B
n sin⎜
⎟.
L
⎝
⎠
⎝ L ⎠
⎛ nπt ⎞
Solve each driven ODE y n” +
ay n' +
by n =
An cos⎜
⎟ , etc. and use superposition to
⎝ L ⎠
find sol’n of original ODE. Useful for analysis of resonant frequency, etc.
2. Defined spaces of real/complex-valued k-times continuously differentiable/ piecewise ktimes cts. diff. functions
C IR t [a,b], C Cl, t[a,b], PC IRt [a,b], PCCl, t[a,b].
b
∫ f(t)g(t)dt
and talked about properties (= axioms for
Hermitian inner product space). Defined f =
<
f , f > , ∂ mean (f , g) = f − g . Mention this
3. Defined the inner product <
f,g >=
a
is different than the uniform metric.
4. Defined orthogonal and orthonormal sequences.
⎧
⎛ nπt ⎞
⎛ nπt ⎞⎫
Checked that {1} ∪ ⎨cos⎜
gives an orthogonal sequence and
⎟ , sin⎜
⎟⎬
L
⎝
⎠
⎝ L ⎠⎭ n = 1,2 ,....
⎩
computed the norms.
5. Posited existence of a Fourier series
∞
f(t) = A0 +
∑A
n
n =1
Concluded A0 =
⎛ nπt ⎞
⎛ nπt ⎞
cos⎜
⎟ + Bn sin⎜
⎟ for f(t)+PC IR[-L,L]
⎝ L ⎠
⎝ L ⎠
1
2L
L
∫
f (t )dt , An =
−L
18.034, Honors Differential Equations
Prof. Jason Starr
1
L
L
L
1
⎛ nπt ⎞
f(t) cos⎜
f(t) sin dt
⎟dt , B
n =
L
L
⎝
⎠
−L
−L
∫
∫
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