## MathSofiya Group Title I'm trying to find a pattern $a_{n+2}=\frac{a_n}{(n+2)(n+1)}$ $n=0:a_2=\frac{a_0}{2!}$ $n=2:a_4=\frac{a_0}{4!}$ but that pattern changes because the next term is not $\frac{a_0}{6!}$ $n=4:a_6=\frac{a_0}{2\cdot 30}=$ one year ago one year ago

1. Algebraic! Group Title

?? $a _{4} = \frac{a _{0} }{ 4! }$

2. TuringTest Group Title

this is the one we did yesterday? wasn't it$a_{n+2}=-a_n\frac{1-n}{(n+1)(n+2)}$?

3. MathSofiya Group Title

no this is new problem $y''=y$

4. TuringTest Group Title

oooooooh

5. MathSofiya Group Title

I'll write out what I have so far

6. TuringTest Group Title

I'll be able to able to help in about 15-20 min

7. MathSofiya Group Title

sure

8. MathSofiya Group Title

$\sum_{n=2}^\infty n(n-1)a_nx^{n-2}-\sum_{n=0}^\infty a_nx^n=0$

9. nipunmalhotra93 Group Title

@MathSofiya how exactly is the pattern changing?

10. TuringTest Group Title

let's establish that we have the right pattern first... I feel like you may have missed something continue plz @MathSofiya so far so good

11. MathSofiya Group Title

$\sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^n-\sum_{n=0}^\infty a_nx^n=0$

12. TuringTest Group Title

so yeah, right pattern

13. MathSofiya Group Title

$\sum_{n=0}^\infty x^n\left[(n+2)(n+1)-a_n\right]=0$

14. TuringTest Group Title

$\sum_{n=0}^\infty x^n\left[(n+2)(n+1)a_{n+2}-a_n\right]=0$

15. MathSofiya Group Title

oops

16. TuringTest Group Title

$a_{n+2}=\frac{a_n}{(n+1)(n+2)}$so you were right, just plug in each n and write out $$everything$$ explicitly, even multiplication by 1

17. MathSofiya Group Title

ok

18. MathSofiya Group Title

$n=0:a_{0+2}=a_2=\frac {a_0}{2}$ $n=1:a_{1+2}=a_3=\frac {a_1}{6}$ $n=2:a_{2+2}=a_4=\frac {a_2}{12}$ $n=3:a_{3+2}=a_5=\frac {a_3}{20}$ $n=4:a_{4+2}=a_6=\frac {a_4}{30}$ $n=5:a_{5+2}=a_7=\frac {a_5}{42}$

19. MathSofiya Group Title

so we established the relationship between $a_2=\frac{a_0}{2}$ but how do I continue this to the next line? how do I change a_2 to a_0?

20. MathSofiya Group Title

ooohh no need to because the next line is going to be zero

21. TuringTest Group Title

you are not writing it out explicitly enough to see the pattern reduce everything to terms of either a_0 or a_1 the two constants we can represent no more simply

22. TuringTest Group Title

now is the next line zero?

23. TuringTest Group Title

how*

24. MathSofiya Group Title

well it was yesterday LOL ....Ok I'll write everything out

25. TuringTest Group Title

$n=0:a_{0+2}=a_2=\frac {a_0}{2\cdot1}$$n=1:a_3=\frac{a_1}{3\cdot2}$write everything this way, with each number on the bottom specified

26. TuringTest Group Title

yesterday we had n-1 in the top, that's why

27. TuringTest Group Title

Haha I'm exercising while you crunch the numbers :P

28. MathSofiya Group Title

$n=0:a_{0+2}=a_2=\frac {a_0}{2}$ $n=1:a_{1+2}=a_3=\frac {a_1}{6}=\frac{a_1}{2\cdot 3}$ $n=2:a_{2+2}=a_4=\frac {a_2}{12}=\frac{a_0}{2\cdot2\cdot3}$ $n=3:a_{3+2}=a_5=\frac {a_3}{20}=\frac{a_3}{10\cdot2}=\frac{a_1}{2\cdot2\cdot2\cdot3\cdot5}$ $n=4:a_{4+2}=a_6=\frac {a_4}{30}=\frac{a_0}{2\cdot2\cdot2\cdot3\cdot5}$ $n=5:a_{5+2}=a_7=\frac {a_5}{42}=\frac{a_1}{2\cdot2\cdot2\cdot3\cdot5\cdot42}$

29. MathSofiya Group Title

n=4 is wrong oh well...but I see a pattern now. How do you workout and do math?

30. TuringTest Group Title

push ups, then run over and check your work real quick :)

31. TuringTest Group Title

n=2 is wrong

32. TuringTest Group Title

12 is not 2x3

33. MathSofiya Group Title

haha! That's what I call commitment!

34. MathSofiya Group Title

give me a second to do the problem on paper. It's kinda tedious to type it...one second

35. TuringTest Group Title

$a_n=\frac{a_n}{(n+1)(n+2)}$$n=0:a_{0+2}=a_2=\frac {a_0}{2\cdot1}$$n=1:a_3=\frac{a_1}{3\cdot2}$$n=2:a_4=\frac {a_2}{4\cdot3}=\frac{a_0}{4\cdot3\cdot2\cdot1}$$n=3:a_5=\frac {a_3}{5\cdot4}=\frac{a_1}{5\cdot4\cdot3\cdot2}$

36. TuringTest Group Title

write out each number in the denom without simplifying every time

37. TuringTest Group Title

I cannot stress that enough to find patterns

38. MathSofiya Group Title

$n=0:a_{0+2}=a_2=\frac {a_0}{2}$ $n=1:a_{1+2}=a_3=\frac {a_1}{6}=\frac{a_1}{3\cdot 2}$ $n=2:a_{2+2}=a_4=\frac {a_2}{12}=\frac{a_0}{4\cdot3\cdot2\cdot1}$ $n=3:a_{3+2}=a_5=\frac {a_3}{20}=\frac{a_3}{10\cdot2}=\frac{a_1}{5\cdot4\cdot3\cdot2\cdot1}$ Ok I'm finally convinced...had to calculate like 5 times that the denominator is actually 120

39. TuringTest Group Title

good, but more importantly what is/are the general pattern(s)?

40. TuringTest Group Title

again, there are two; one for even n and one for odd n

41. MathSofiya Group Title

sorry internet is super slow. I'm gonna restart my computer brb

42. MathSofiya Group Title

ok I'm back

43. TuringTest Group Title

I'm getting in the shower soon... so what are the two constants that every other constant $$a_n$$ can be written in terms of ?

44. MathSofiya Group Title

I wanna say $(n+2)!$ and then sub in n=2k

45. MathSofiya Group Title

wait...you go shower, and let me work on this

46. TuringTest Group Title

no, and that would only be evens, and needs a base constant like $$a_0$$ write it as$a_n=?$and remember this time we do have odd terms

47. MathSofiya Group Title

allow me to decipher it...

48. MathSofiya Group Title

The denominator for both even and odds is the following...still working on the numerator $\frac{?}{(n+2)!}$

49. TuringTest Group Title

look at the difference between constants with an even and odd subscript what is the main difference you see>?

50. MathSofiya Group Title

one will always be zero and the other will always be 1. I have come to that conclusion, but now I have to find a way to write it mathematically...I'm gonna cheat and look at yesterdays work

51. TuringTest Group Title

no wait, you can do it... let me give another hint...

52. MathSofiya Group Title

ok

53. TuringTest Group Title

odd are reducible to being in terms of the constant a_1 evens are reducible to being in terms of a_0 and for the denominator for any constant $$a_n$$ what is it? look at the denom for a_2, a_3, a_4, what are they in terms of n ?

54. TuringTest Group Title

don't get confused with the n=(whatever) thing I mean what is the relation between the n in the subscript $$a_n$$ and the number in the denominator?

55. MathSofiya Group Title

hold on I didn't write anything yet

56. MathSofiya Group Title

57. TuringTest Group Title

no rush, still haven't showered yet

58. MathSofiya Group Title

When the numerator is $a_0$ the denominator is even When the numerator is $a_1$ the denominator is odd

59. MathSofiya Group Title

by odd I mean 5! 3! and so on

60. TuringTest Group Title

exactly, and all even numbers can be written how?

61. MathSofiya Group Title

2k

62. TuringTest Group Title

and odd?

63. MathSofiya Group Title

n=2k

64. MathSofiya Group Title

n=2k+1

65. MathSofiya Group Title

$a_{2k}$ and $a_{2k+1}$

66. TuringTest Group Title

so then what is our formulas? we need one for all $$a_{2k}$$ and another for all $$a_2k+1}$$

67. TuringTest Group Title

right, we need a formula for each one.. what are they ? that's probably the hardest part of the problem in my opinion is finding the pattern

68. MathSofiya Group Title

I would like to put a_{2k} in the numerator and (n+2) in the numerator, but that wouldn't be right. So I have write it in one formula to account for both the even and odd?

69. MathSofiya Group Title

I can't write two separate formulas correct? I've gotta write one?

70. TuringTest Group Title

everything should be written in terms of either a_0 and k, or a_1 and k n is now out of the poicture completely

71. TuringTest Group Title

picture*

72. MathSofiya Group Title

gtg...I know this is the last part, and shouldn't take too long, but I gtg to the gym. i'll be back at 7:30 8pm

73. MathSofiya Group Title

ok bye

74. TuringTest Group Title

cool, health makes one think more clearly I believe see ya

75. MathSofiya Group Title

LOL I hope soo! see ya

76. MathSofiya Group Title

where is Alan Turing?

77. MathSofiya Group Title

I'm reviewing the previous problems we did...and there might be some errors. I not convinced with the way that the final answer was written in k's

78. TuringTest Group Title

Allow me to restore your faith, though there are other manners of making the notation I'm sure... Paul to the rescue again: http://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx Notice that since the constants become different depending on even-odd subscripts for many answers we divide the subscripts in the recursion formula for them into terms of 2k and 2k+1. This particular problem you are doing is almost identical to example 1 one this page, with the exception of the minus between y'' and y exchanged for a plus (reading this example you will probably discover the pattern I was asking you to look for). Notice that it is in fact essential that we change the terms from n to k, since we can get no single summation formula in terms of n since the coefficients depend on whether n is odd n. This is not true for all problems with series solutions, but when we need to distinguish between even and odd n the only way to do this is to put them as two separate series in terms of k. Read example 1 carefully to understand the intricacies of the process, and the why we put n in terms of k. Sorry I never returned last night, I was teaching political philosophy. Look forward to finishing the problem with you today if possible. Tag me when you are ready to continue :)

79. MathSofiya Group Title

I'll be back in like 10 mins, gotta start laundry

80. TuringTest Group Title

right-o

81. MathSofiya Group Title

I can't find my laundry card! Ok so I was trying to rewrite the first problem, and got to the part where 1. all even n#'s =0, or all odd "a" subscripts=0 2. We've gotten a pattern that we called $a_n=\frac{a_0}{2^n(n!)}$ but if we call them "n" for now wouldn't that conflict because Let's say for n=5 $\frac{a_0}{2^3(3!)}$ which is not true (I'm referring to y'=xy, the first problem)

82. MathSofiya Group Title

Well it's what we called the pattern initially before writing it in terms of k's

83. MathSofiya Group Title

ok found the laundry card brb

84. TuringTest Group Title

The pattern is not in terms of n like that, if it were it would be$a_n=\frac{a_0}{2^{n/2}(\frac n2!)}$for even n subscripts of a, and$a_n=0$ for all odd n subscripts of a. I think a major problem here is that we are getting confused with the n in the $$n=...$$ that we are using to plug in to find the particular constant $$a_n$$, and the n in the $$a_n$$ in the constant. Those two n's are not equal! What exactly is your argument that in the last problem you are referring too we do no not have that$n=5:a_6=\frac{a_4}6=\frac{a_0}{2\cdot4\cdot6}=\frac{a_0}{2^3(3!)}$??

85. MathSofiya Group Title

Ok that makes sense. I thought the n's were the same. So the n's are just a variable to hold place until we change the argument to k's I guess. wait, what do you mean by the last sentence? so the n=5 statement above is not accurate?

86. TuringTest Group Title

yes it is accurate, plugging in n=5 into the recursion formula$a_{n+1}=\frac{a_{n-1}}{n+1}$gives$n=5:a_6=\frac{a_4}6=\frac{a_0}{2\cdot4\cdot6}=\frac{a_0}{2^3(3!)}$the $$n=...$$ is just what we use to get an expression relating two other constants $$a_{n+ p}$$and $$a_{n+ q}$$ where p and q are some number (in this case above p=-1 and q=1) so the n we plug in to test is $$not$$ the same as the n in the subscripts, because the subscripts in our recursion relation are usually not just $$a_n$$ (the same n as we are plugging in for the test) but really $$a_{n+p}$$

87. TuringTest Group Title

We want our expression for each constant in terms of its $$subscript$$, not the $$n=...$$ we use to test the relation, hence we divide the subscripts into even and odd to make the pattern more clear. Once we notice that by doing that we can get meaningful expressions for each constant a in terms of the subscript (however we write) that is what we use...

88. TuringTest Group Title

in this case by dividing the subscripts into those of the form 2k and those of the form 2k+1 we get a new way to write each constant in terms of k

89. TuringTest Group Title

...and whatever base constant we can't determine, like $$a_0$$ which we do not know since we are not given the initial conditions

90. MathSofiya Group Title

ok that makes more sense now. Let's see if I can apply it to this problem now.

91. TuringTest Group Title

go for it, I'm gonna get a few empanadas for breakfast :) (sortof like Mexican hot-pockets :) brb

92. MathSofiya Group Title

sounds good

93. MathSofiya Group Title

Let's see here.... $a_2=\frac{a_0}{2!}$ $a_3=\frac{a_1}{3!}$ $a_4=\frac{a_0}{4!}$ $a_5=\frac{a_1}{5!}$ $a_2k=\frac{a_0}{k!}$ my way of saying if the subscript is even...

94. MathSofiya Group Title

$a_{2k+1}=\frac{a_1}{k!}$

95. TuringTest Group Title

is it? what is$$a_3$$ ?

96. MathSofiya Group Title

oh my 2k+1=3 2k=2 k=1

97. MathSofiya Group Title

yeah I'm wrong

98. MathSofiya Group Title

3k?

99. MathSofiya Group Title

your patients with me is truly astonishing. I would have strangled the student.

100. TuringTest Group Title

haha, but I haven't done this in a long time either, so it's a good review for me too.

101. MathSofiya Group Title

*patience

102. TuringTest Group Title

$a_n=\frac{a_n}{(n+1)(n+2)}$$n=0:a_2=\frac {a_0}{2}=\frac{a_0}{2!}$$n=1:a_3=\frac {a_1}{6}=\frac{a_1}{3\cdot 2}=\frac{a_1}{3!}$$n=2:a_4=\frac {a_2}{12}=\frac{a_0}{4\cdot3\cdot2\cdot1}=\frac{a_0}{4!}$$n=3:a_5=\frac {a_3}{20}=\frac{a_3}{10\cdot2}=\frac{a_1}{5\cdot4\cdot3\cdot2\cdot1}=\frac{a_1}{5!}$now hopefully you have already noticed that the denominator on each $$a_n$$ is $$n!$$$a_n=\frac{a_0}{n!}$if n is even, and$a_n=\frac{a_1}{n!}$if n is odd given that, simply rewrite the subscripts as 2k or 2k+1 depending on whether they are even or odd, and then sub in for n in the general expressions

103. MathSofiya Group Title

yep, I honestly saw that relationship all along, I just didn't know how to express that in k's $a_{2k}=\frac{a_0}{2k}$ $a_{2k+1}=\frac{a_0}{2k+1}$ sorry I kinda feel silly now

104. MathSofiya Group Title

way easy

105. TuringTest Group Title

you forgot the factorials ;)

106. MathSofiya Group Title

!!!!!!!!!!!! there

107. MathSofiya Group Title

LOL ok let me rewrite them

108. TuringTest Group Title

and you forgot that for odd subscripts it is a_1 in the numerator

109. MathSofiya Group Title

$a_{2k}=\frac{a_0}{(2k)!}$ \a_{2k+1}=\frac{a_1}{2k+1}[\] I didn't forgot...I just checking if you were paying attention :P

110. MathSofiya Group Title

$a_{2k+1}=\frac{a_1}{2k+1}$

111. MathSofiya Group Title

oh my

112. TuringTest Group Title

haha :P

113. MathSofiya Group Title

$a_{2k+1}=\frac{a_1}{(2k+1)!}$

114. MathSofiya Group Title

happy?

115. MathSofiya Group Title

oh wait there is more...

116. TuringTest Group Title

Much better :) I wanna make a subtle point about the possible values of k we are talking about, but I will save it till the end so as not to confuse you.

117. MathSofiya Group Title

$y=\sum_{n=0}^\infty a_nx^n=a_0\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}$ $y=\sum_{n=0}^\infty a_nx^n=a_1\sum_{k=0}^\infty\frac{x^{2k+1}}{(2k+1)!}$

118. TuringTest Group Title

you've got the right idea, but we need to use superposition to get the full answer

119. TuringTest Group Title

if we accept only one answer or the other we are excluding either all the even or odd exponents, so how can we fix that?

120. MathSofiya Group Title

put them into one formula somehow

121. MathSofiya Group Title

oh y and c_1 and so on and so forth...hold on

122. TuringTest Group Title

superposition...

123. MathSofiya Group Title

124. TuringTest Group Title

yep :) c1=a0 c2=a1

125. MathSofiya Group Title

$c_1\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}+c_2\sum_{k=0}^\infty\frac{x^{2k+1}}{(2k+1)!}=0$ huh?

126. MathSofiya Group Title

gosh the sums probably shouldn't be there...

127. TuringTest Group Title

no I was just saying that the c1 you are used to is called a0 in this case, so...

128. TuringTest Group Title

$y=a_0\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}+a_1\sum_{k=0}^\infty\frac{x^{2k+1}}{(2k+1)!}=0$

129. TuringTest Group Title

wait, why =0 ?

130. TuringTest Group Title

$y=a_0\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}+a_1\sum_{k=0}^\infty\frac{x^{2k+1}}{(2k+1)!}$

131. MathSofiya Group Title

I just looked at the example in the book Uhm Ohhh,,, but they don't have to be written in c_1 and c_2 right?

132. MathSofiya Group Title

can I put my laundry in the dryer?

133. MathSofiya Group Title

brb

134. MathSofiya Group Title

2 mins

135. TuringTest Group Title

nah, constants are constants, you can call them whatever you want but I just wanted to keep our notation consistent, and since they are equivalent to c1 and c2 (constants that depend on the initial conditions) and yes, you have my permission to do your laundry :P

136. MathSofiya Group Title

ok I'm back

137. MathSofiya Group Title

yep makes sense

138. TuringTest Group Title

so review what we did, look again at example one one here: http://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx and tell me if you have any questions

139. TuringTest Group Title

example 1 on*

140. MathSofiya Group Title

I read it, I don't think I have any questions. 22 hrs later I finally understand it (hopefully I'll be able to recreate it)

141. TuringTest Group Title

awesome! there is, as I said, a subtle point to be made about the possible values of k (as paul mentions) but since it really doesn't affect this problem I'll leave it for you to read, or we'll cross that bridge when we come to it :)

142. MathSofiya Group Title

sounds good...I'll do 2 problems on my own from beginning to end and we'll see if I make error or still struggle on some points here and there...

143. TuringTest Group Title

I do encourage you to read the whole page I linked you to though, to see some of the differences encountered in other problems. Good luck!

144. MathSofiya Group Title

145. TuringTest Group Title

nope, bought 'em at the corner I only cook hamburgers and pasta

146. MathSofiya Group Title

what? you've been in Mexica for a little while now! Gotta learn to cook like the locals!

147. TuringTest Group Title

well I make quesadillas, I guess that's a start lol

148. MathSofiya Group Title

yep, work your way up from here :P

149. TuringTest Group Title

:)

150. mahmit2012 Group Title

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151. mahmit2012 Group Title

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152. mahmit2012 Group Title

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153. mahmit2012 Group Title

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