Question

Find two power series solutions of the given differential equation about the ordinary point x=0: y''-xy=0

Answer 1

fourier series

Answer 2

.....

Answer 3

\[y=\sum_{n=0}^\infty c_nx^n\]

Answer 4

\[y'=\sum_{n=1}^\infty c_nnx^{n-1}\]

Answer 5

\[y''=\sum_{n=2}^\infty c_n(n)(n-1)x^{n-2}\]

Answer 6

oh I need a review on this
bookmark*

Answer 7

i'm pretty sure i have it down so you can come check it after

\[y''-xy=\sum_{n=2}^\infty c_nn(n-1)x^{n-2}-x\sum_{n=0}^\infty c_nx^n\]

Answer 8

\[\sum_{n=2}^\infty c_nn(n-1)x^{n-2}-\sum_{n=0}^\infty c_nx^{n+1}\]

Answer 9

\[k=n-2\]
\[k+2=n\]

\[\sum_{k=0}^\infty c_{k+2}(k+2)(k+2-1)x^k-\sum_{n=0}^\infty c_nx^{n+1}=0\]

Answer 10

\[\sum_{k=0}^\infty c_{k+2}(k+2)(k+1)x^k-\sum_{k=1}^\infty c_{k-1}x^k\]

where

\[k=n+1\]
\[k-1=n\]

Answer 11

pull out one term k=0 of first

\[2c_2+\sum_{k=1}^\infty c_{k+2}(k+2)(k+1)x^k-\sum_{k=1}^\infty c_{k-1}x^k\]

Answer 12

\[2c_2+\sum_{k=1}^\infty [c_{k+2}(k+2)(k+1)-c_{k-1}]x^k\]

Answer 13

since identity 0=0 the sum and

\[2c_2=0\]
\[c_2=0\]

Answer 14

\[c_{k+2}(k+2)(k+1)=c_{k-1}\]

\[c_{k+2}=\frac{c_{k-1}}{(k+2)(k+1)}\]

Answer 15

for k=1,2,3,4,5,6......

Answer 16

k=1

\[c_3=\frac{c_0}{3*2}\]
k=2

\[c_4=\frac{c_1}{4*3}\]

for k =3

\[c_5=\frac{c_2}{constant}\] =\[c_2=0\]

k=4

\[c_6=\frac{c_3}{6*5)}=\frac{c_0}{6*5*3*2}\]

Answer 17

k=5

\[c_7=\frac{c_4}{7*6}=\frac{c_1}{7*6*4*3}\]

Answer 18

\[c_8=c_5=0\]

Answer 19

@TuringTest look correct so far?

Answer 20

like I said: *bookmark
I want you to remind me; it's been a while...
maybe @experimentX can verify better than me

Answer 21

jeez ... i forgot all those stuffs. i need to review it myself.

Answer 22

hm... @Zarkon care to verify a DE series solution?

Answer 23

Is zarkon the computer on lol

Answer 24

these standard equations power solutions look intimidating ...

Answer 25

\[y=c_0+c_1x+0+\frac{c_0}{3*2}x^3+\frac{c_1}{4*3}x^4+0+\frac{c_0}{2*3*5*6}x^6\]

\[+\frac{c_1}{3*4*6*7}x^7+0......\]

Answer 26

\[y_1(x)=1+\frac{1}{2*3}x^3+\frac{1}{2*3*5*6}x^6...\]

\[y_2(x)=x+\frac{1}{3*4}x^4+\frac{1}{3*4*6*7}x^7.......\]

Answer 27

what kind of function is this
http://www.wolframalpha.com/input/?i=y%27%27+-+xy%3D0

Answer 28

Using Maple I found this solution
\[ C0+C1*x+(1/6)*C0*x^3+(1/12)*C1*x^4+\\ (1/180)*C0*x^6+(1/504)*C1*x^7+O(x^8) \]

Answer 29

thats what i have

Answer 30

only i left it not multiplied so i can creat a summation

Answer 31

well, i guess you are right ... i never liked power series solution you know!!