As before a and a 1 are arbitrary with y 0 a and y 0 a 1 It follows that a 2 16

As before a and a 1 are arbitrary with y 0 a and y 0

This preview shows page 23 - 24 out of 24 pages.

As before, a 0 and a 1 are arbitrary with y (0) = a 0 and y 0 (0) = a 1 It follows that a 2 = - 16 2 a 0 = - 8 a 0 , a 4 = 4 - 16 4 · 3 a 2 = 8 a 0 , a 6 = 0 = a 8 = ... = a 2 n and a 3 = - 15 3 · 2 a 1 = - 5 2 a 1 , a 5 = - 7 5 · 4 a 3 = 7 8 a 1 , a 7 = 9 7 · 6 a 5 = 3 16 a 1 , ... Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (23/24)
Image of page 23

Subscribe to view the full document.

Introduction Series Solutions of Differential Equations Airy’s Equation Chebyshev’s Equation Chebyshev’s Equation 4 Chebyshev’s Equation with α = 4 : From the recurrence relation , we see that the even series terminates after x 4 , leaving a 4 th order polynomial solution. The general solution becomes: y ( x ) = a 0 ( 1 - 8 x 2 + 8 x 4 ) + a 1 x - 5 2 x 3 + 7 8 x 5 + 3 16 x 7 + ... y ( x ) = a 0 ( 1 - 8 x 2 + 8 x 4 ) + a 1 x + X n =1 [(2 n - 1) 2 - 16][(2 n - 3) 2 - 16] · · · · · (3 2 - 16)(1 - 16) (2 n + 1)! x 2 n +1 ! More generally, it is not hard to see that for any α an integer, the Chebyshev’s Equation results in one solution being a polynomial of order α (only odd or even terms). The other solution is an infinite series. The polynomial solution converges for all x , while the infinite series solution converges for | x | < 1. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (24/24)
Image of page 24
  • Fall '08
  • staff

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

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes