Morework_13 - Morework Solutions Math 213 Spring 2008 Week 13 5.1#8 For which values of does the boundary value problem y 2y(1)y = 0 y(0 = 0 y(1 =

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Morework Solutions Math 213 – Spring 2008 Week 13 5.1#8: For which values of λ does the boundary value problem y 00 - 2 y 0 + (1 + λ ) y = 0 , y (0) = 0 , y (1) = 0 have a solution. As usual, we look at the three separate cases: Case λ > 0 . Then the characteristic polynomial of the equation is r 2 - 2 r + (1 + λ ) = 0 and the roots are r = 2 ± p 4 - 4(1 + λ ) 2 = 2 ± - 4 λ 2 = 1 ± where μ = λ . So the general solution will have the form y = c 1 e x cos( μx ) + c 2 e x sin( μx ) . Our boundary conditions then give the following equations: 0 = c 1 , 0 = c 1 e cos( μ ) + c 2 e sin( μ ) = c 2 e sin( μ ) . So we will have a nontrivial solution exactly when sin( μ ) = 0, which hap- pens when μ = π, 2 π, 3 π,. .. (Note that μ > 0, so 0 , - π, - 2 π,. .. must be discarded.) Therefore, the equation has nontrivial solutions when λ = π 2 , 4 π 2 , 9 π 2 ,... . Case λ = 0 . The characteristic polynomial of the equation is r 2 - 2 r + 1 = ( r - 1) 2 = 0. So the general solution will have the form y = c 1 e x + c 2 xe x . 1
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Our boundary conditions then give the following equations: 0 = c 1 , 0 = c 1 e + c 2 e = c 2 e. We conclude that
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This note was uploaded on 05/09/2008 for the course MATH 2130 taught by Professor Dorais during the Spring '08 term at Cornell University (Engineering School).

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Morework_13 - Morework Solutions Math 213 Spring 2008 Week 13 5.1#8 For which values of does the boundary value problem y 2y(1)y = 0 y(0 = 0 y(1 =

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