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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Image of page 1

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

View Full Document Right Arrow Icon
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-
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.
  • Spring '11
  • NormanKatz
  • Eigenvalue, eigenvector and eigenspace, wave equation, Eigenfunction, infinite sum, Vibrations of a circular drum

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern