Math450hw7sol - MATH 450 Section 002 Homework 7 Solutions...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 450 Section 002 Homework 7 Solutions Winter 2009 Section 20.2, p. 1067, #15. (a) Let u ( x,y ) = f 2 x 2 + U ( x,y ), then U ( x,y ) satisfies U xx + U yy = 0 , U (0 ,y ) = 0 , U ( a,y ) =- f 2 a 2 , U ( x, 0) = U ( x,b ) =- f 2 x 2 . Seeking U ( x,y ) = X ( x ) Y ( y ), then X 00 ( x ) Y ( y ) + X ( x ) Y 00 ( y ) = 0 . Therefore X 00 ( x ) X ( x ) =- Y 00 ( y ) Y ( y ) =- 2 . This implies X ( x ) = A + Bx, if = 0 , C cos x + D sin x, if 6 = 0 , and Y ( y ) = E + Fy, if = 0 , Ge y + He- y , if 6 = 0 . so u ( x,y ) = ( A + Bx )( E + Fy ) + ( C cos x + D sin x )( Ge y + He- y ) . Now lets apply the boundary conditions. First, u (0 ,y ) = 0 = A ( E + Fy ) + C ( Ge y + He- y ) , 1 so A = C = 0. Thus u ( x,y ) = Ix + Jxy + ( Pe y + Qe- y )sin x. Second, u ( a,y ) =- fa 2 2 = Ia + Jay + ( Pe y + Qe- y )sin a, therefore, I =- fa 2 , J = 0 , sin a = 0 . It follows from the above equations that = n a , n = 1 , 2 , 3 , . Thus u ( x,y ) =- fa 2 x + X n =1 ( P n e n a y + Q n e- n a y )sin n a x. Furthermore, one has (1) u ( x, 0) =- fx 2 2 =- fa 2 x + X n =1 ( P n + Q n )sin n a x, so P n + Q n = 2 a Z a ( fa 2 x- f 2 x 2 )sin n a xdx. Finally, u ( x,b ) =- fx 2 2 = fa 2 x + X n =1 ( P n e nb a + Q n e- nb a )sin nx a dx, 2 so (2) P n e nb a + Q n e- nb a = 2 a Z a ( fa 2 x- f 2 x 2 )sin nx a dx. Solve the system (1) and (2) for P n and Q n , one has P n = e- nb a- 1 e- nb a- e nb a 2 a Z a ( fax 2- f 2 x 2 )sin nx a dx, Q n = e nb a- 1 e nb a- e- nb a 2 a Z a ( fax 2- f 2 x 2 )sin nx a dx....
View Full Document

This note was uploaded on 04/24/2010 for the course MATH 425 taught by Professor K during the Spring '10 term at University of Michigan-Dearborn.

Page1 / 10

Math450hw7sol - MATH 450 Section 002 Homework 7 Solutions...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online