mid2sols - Name: 110 MIDTERM 2 Some potentially useful...

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Unformatted text preview: Name: 110 MIDTERM 2 Some potentially useful formulae: ( i ) cosh x = 1 2 ( e x + e − x ). ( ii ) The Fourier coefficients of f ( x ) for the eigenfunctions cos( nπx/ℓ ) and sin( nπx/ℓ ) on the interval ( − ℓ, ℓ ) are A n = 1 ℓ integraldisplay ℓ − ℓ f ( x ) cos nπx ℓ d x B n = 1 ℓ integraldisplay ℓ − ℓ f ( x ) sin nπx ℓ d x. In this case the Fourier series for f ( x ) is 1 2 A + ∞ summationdisplay n =1 ( A n cos nπx ℓ + B n sin nπx ℓ ) . ( iii ) The Fourier coefficients of f ( x ) for the eigenfunctions e inπx/ℓ on the interval ( − ℓ, ℓ ) are c n = 1 2 ℓ integraldisplay ℓ − ℓ f ( x ) e − inπx/ℓ d x. In this case the Fourier series for f ( x ) is ∞ summationdisplay n = −∞ c n e inπx/ℓ . score 1 2 3 total 1 Name: 110 MIDTERM 2 1. Let φ ( x ) = sin x . a. [10 points] Solve u t = u xx for 0 < x < π and 0 < t < ∞ , with boundary conditions u (0 , t ) = 0 = u ( π, t ) and initial condition u ( x, 0) = φ ( x )....
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mid2sols - Name: 110 MIDTERM 2 Some potentially useful...

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