Differential Equations Lecture Work Solutions 44

# Differential Equations Lecture Work Solutions 44 - 6 4 2 0...

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-8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 -L L -8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 -L L Figure 29: Sketch of f ( x ) and its periodic extension for problem 3 3. f ( x ) = 1 x < L/ 6 3 L/ 6 < x < L/ 2 0 x > L/ 2 Fourier cosine series coeﬃcients: a 0 = 2 L L/ 6 0 dx + L/ 2 L/ 6 3 dx = 2 L L 6 + 3 L 2 L 6 = 2 L L 6 + 3 2 L 6 = 2 L · 7 6 L = 7 3 a n = 2 L L/ 6 0 cos n π L x dx + 3 L/ 2 L/ 6 cos n π L x dx = 2 L sin n π L x n π L | L/ 6 0 + 3 sin n π L x n π L | L/ 2 L/ 6 = 2 L sin n π 6 n π L + 3
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