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Differential Equations Lecture Work Solutions 45

Differential Equations Lecture Work Solutions 45 - 5 0 L L...

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-10 -5 0 5 10 -5 0 5 -L L -10 -5 0 5 10 -5 0 5 -L L Figure 30: Sketch of the even extension of f ( x ) and its periodic extension for problem 3 4. x [ L, L ] y [ a, b ] then y = a + b 2 + b a 2 L x ( ) is the transformation required (Note that if x = L then y = a and if x = L then y = b ) g ( y ) is periodic of period b a g ( y ) = G ( x ) = a 0 2 + n =1 ( a n cos n π L x + b n sin n π L x ) a n = 1 L L L G ( x ) cos n π L x dx solving (*) for x yields x = 2 L b a y a + b 2 dx = 2 L
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