{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

homework 3 solutions addon

homework 3 solutions addon - Homework 3 part 2 Solutions...

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

View Full Document Right Arrow Icon
Homework 3, part 2: Solutions and remarks on selected problems Greenberg, 4.3, 6(a). The equation is 2 x 2 y 00 + xy 0 + x 4 y = 0. (We have multiplied through by an additional factor of x to simplify bookkeeping.) Substitution of 0 a n x r + n in the equation leads to X n =0 [ F ( r + n ) a n + a n - 4 ] x r + n = 0 , (1) where as usual, a - k = 0 for integers k > 0, and where F ( r ) = 2 r ( r - 1) + r = r (2 r - 1). The coefficient corresponding to n = 0 in (1) is set to 0 by choosing r to satisfy the indicial equation F ( r ) = r (2 r - 1) = 0. The two roots are r 1 = 1 / 2 and r 2 = 0. To find a solution corresponding to r 1 set r = 1 / 2 in (1), and set the coefficients for n 1 equal to zero, to get the recursion equation a n = - 1 F ( n +(1 / 2)) a n - 4 = - 1 2 n ( n + 1 / 2) a n - 1 = 0 , n 1 . (2) Since a n - 4 = 0 if n < 4, this recursion relation implies a 1 = a 2 = a 3 = 0. Next, taking n = 4 in (2), a 4 = - a 0 / 36. Then a 5 = a 6 = a 7 = 0 because by the recursion formula they are multiples of a 1 , a 2 , and a 3 respectively. Next, for n = 8, a 8 = - a 4 / (136) = a 0 / (36 · 136) = a 0 / 4896. It is clear that only every fourth coefficient will be non-zero. Setting a 0 = 1, y 1 ( x ) = x 1 / 2 1 - x 4 36 + x 8 4896 + a 12 x 12 + · · · ! . The recursion relations can be solved explicitly using the Gamma function to get y 1 ( x ) = x 1 / 2 X k =0 ( - 1) k Γ(9 / 8) (32) k k !Γ( k +(9 / 8)) x 4 k . To find a solution corresponding to r 2 set r = 0 in (1), and set the coefficients for n 1 equal to zero, to get the recursion equation a n = - 1 F ( n ) a n - 4 = - 1 2 n ( n - 1 / 2) a n - 1 = 0 , n 1 . (3) As before, since a n - 4 = 0 if n < 4, this recursion relation implies a 1 = a 2 = a 3 = 0. Next, taking n = 4 in (3), a 4 = - a 0 / 28. Again, a 5 = a 6 = a 7 = 0 because by the recursion formula they are multiples of a 1 , a 2 , and a 3 respectively, and, continuing, all
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ 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