# C if y j ? n x justify the equality y a 0 and get y a

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You will need to use integration by parts on the second term. c) If y = J 0 ( λ n x ), justify the equality y ( a ) = 0 and get [ y ( a )] 2 = 2 λ 2 n a 2 integraldisplay a 0 x [ y ( x )] 2 dx d) Use part c) and the fact that J 0 ( x ) = J 1 ( x ) to show that bardbl J 0 ( λ n x ) bardbl 2 = a 2 J 1 ( α n ) 2 / 2 (keep in mind that the norm is worked out on the interval [0 , a ] and that the weight function is w ( x ) = x ). (Note: this is a modified version of problem 36 in section 4.8, if you want to read the original.) 4) (10 points) As on the transcendental ODE handout, let E ( x ) be the unique solution of y = y , y (0) = 1, and let L ( x ) be the unique solution of xy = 1, L (1) = 0. As with the handout, you must do this problem using the ODE information ONLY. a) Show that for all x , L ( E ( x )) = x . b) Show EITHER E ( x 1 + x 2 ) = E ( x 1 ) E ( x 2 ) OR L ( x 1 x 2 ) = L ( x 1 ) + L ( x 2 ). You don’t need to do both of these, and I will only grade one (at random) if you choose to submit them both.

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5) (10 points) a) Use the eigenfunction method to solve the following PDE on the square 0 x 1, 0 y 1, with boundary conditions u ( x, 0) = u ( x, 1) = u (0 , y ) = u (1 , y ) = 0. (Hint: Your answer will NOT be an infinite sum.) 4 2 u ∂x 2 + 5 2 u ∂y 2 = sin(3 πx ) sin(2 πy ) b) Same set up, different f ( x, y ). Use the eigenfunction method to solve the following PDE on the square 0 x 1, 0 y 1, with boundary conditions u ( x, 0) = u ( x, 1) = u (0 , y ) = u (1 , y ) = 0. (Hint: Now you’ll have an infinite sum.) 2 u ∂x 2 + 2 u ∂y 2 = xy 6) (10 points) Find all eigenvalues and corresponding eigenfunctions for the regular Sturm-
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• Fall '11
• NormanKatz
• Eigenvalue, eigenvector and eigenspace, wave equation, Eigenfunction, infinite sum, Vibrations of a circular drum

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