{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

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

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern