midterm2-soln

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 8

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online