midterm2-soln - Midterm Exam II Solution November 2 2011 1...

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Unformatted text preview: Midterm Exam II Solution November 2, 2011 1 Show that the following function u ( x, t ) = 1 √ 4 πνt e- x 2 / 4 νt satisfy the equation u t = νu xx . Solution: u t = 1 √ 4 πν [(- 1 2 t- 3 2 ) e- x 2 / 4 νt + t- 1 2 e- x 2 / 4 νt · (- x 2 4 ν )(- t- 2 )] =- 1 √ 4 πν ( 1 2 t- 3 2- x 2 4 ν t- 5 2 ) e- x 2 / 4 νt u x = 1 √ 4 πνt e- x 2 / 4 νt · (- 2 x 4 νt ) =- 1 √ 4 πνt · x 2 νt e- x 2 / 4 νt u xx =- 1 √ 4 πνt [ 1 2 νt e- x 2 / 4 νt + x 2 νt e- x 2 / 4 νt · (- 2 x 4 νt )] =- 1 √ 4 πνt ( 1 2 νt- x 2 4 ν 2 t 2 ) e- x 2 / 4 νt =- 1 √ 4 πν ( 1 2 ν t- 3 2- x 2 4 ν 2 t- 5 2 ) e- x 2 / 4 νt . Then it is obviously that u t- νu xx = 0 , or u t = νu xx . 1 2 Evaluate the integral ˆ ∞-∞ e- x 2 +2 βx dx Solution: ˆ ∞-∞ e- x 2 +2 βx dx = ˆ ∞-∞ e- ( x- β ) 2 + β 2 dx = e β 2 ˆ ∞-∞ e- ( x- β ) 2 dx = e β 2 ˆ ∞-∞ e- x 2 dx 4 = e β 2 I. I 2 = ( ˆ ∞-∞ e- x 2 dx )( ˆ ∞-∞ e- x 2 dx ) = ( ˆ ∞-∞ e- x 2 dx )( ˆ ∞-∞ e- y 2 dy ) = ˆ ∞-∞ ˆ ∞-∞ e- x 2- y 2 dxdy x = ρ cos θ,y = ρ sin...
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midterm2-soln - Midterm Exam II Solution November 2 2011 1...

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