hw_3_26_27_09

hw_3_26_27_09 - MA 221 Homework Solutions Due date: March...

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MA 221 Homework Solutions Due date : March 26 / 27 , 2009 Page 671 Section 11.2 Problems 1, 3 ̄ ,5 ̄ ,6 ̄ , 13, 1 ̄ 5 ̄ ,1 ̄ 7 ̄ and 1 ̄ 9 ̄ (Underlined problems are to be handed in) In problems 1, 3, 5 and 6 determine the solutions, if any, to the given boundary problem. 1.) y ′′ 2 y 26 y 0 y 0 1, y e r 2 2 r 26 0 r 1 5 i Therefore y x c 1 e x cos5 x c 2 e x sin5 x The BCs imply y 0 c 1 1 y c 1 e e c 1 e e c 1 1 Thus y x e x x c 2 e x x Where C 2 is arbitrary 3 ̄ .) y ′′ 4 y 13 y 0 y 0 0, y 0 r 2 4 r 13 0 r 2 3 i Therefore y x c 1 e 2 t sin3 t c 2 e 2 t cos3 t The BCs imply y 0 c 2 0 y c 1 e 2 0 For all c 1 . Thus y c 1 e 2 t t 1
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3.) y ′′ 4 y 13 y 0 y 0 0, y 0 r 2 4 r 13 0 r 2 3 i Therefore y x c 1 e 2 t sin3 t c 2 e 2 t cos3 t The BCs imply y 0 c 2 0 y c 1 e 2 0 For all c 1 . Thus y c 1 e 2 t t 3 ̄ .) y ′′ 4 y 13 y 0 y 0 0, y 0 r 2 4 r 13 0 r 2 3 i Therefore y x c 1 e 2 t t c 2 e 2 t t The BCs imply y 0 c 2 0 y c 1 e 2 0 For all c 1 . Thus y c 1 e 2 t t 5 ̄ .) y ′′ y sin2 xy 0 y 2 y 0 y 2 2
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r 2 1 0 r  i 0, 1 y h t c 1 e 0 cos x c 2 e 0 sin x y h t c 1 cos x c 2 sin x y p A sin2 x B cos2 x y p 2 A x 2 B x y p ′′ 4 A x 4 B x 4 A x 4 B x A x B x x 3 A x 3 B x x 3 A 1, 3 B 0 A 1/3, B 0 y p 1/3 x y x c 1 cos x c 2 sin x 1/3 x where c 1 and c 2 are the real numbers y 0 c 1 and y 2 c 1 so the condition y 0 y implies nothing. y x c 1 sin x c 2 cos x 2 3 x Thus y 0 c 2 2 3 and y 2 c 2 2 3 so the condition y 0 y 2 also implies nothing. Thus y x c 1 cos x c 2 sin x 1/3 x 6 ̄ .) y ′′ y x y 0 3, y 1 2 e e 1 1 r 2 1 0 r 1 y h c 1 e x c 2 e x y p Ax B y p A y p ′′ 0 0 Ax B x A 1, B 0 y p x Substituting 3
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y 0 3, y 1 2 e e 1 1 c 1 2, c 2 1 y 2 e x e x x In Problems 13, 15, 17 and 19, find all the real eigenvalues and eigenfunctions for the given eigenvalue problem.
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This note was uploaded on 09/22/2009 for the course MA MA221 taught by Professor Levine during the Spring '09 term at Stevens.

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hw_3_26_27_09 - MA 221 Homework Solutions Due date: March...

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