## Outkast3r09 3 years ago Find two power series solutions of the given differential equation about the ordinary point x=0: y''-xy=0

1. 91

fourier series

2. Outkast3r09

.....

3. Outkast3r09

$y=\sum_{n=0}^\infty c_nx^n$

4. Outkast3r09

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

5. Outkast3r09

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

6. TuringTest

oh I need a review on this bookmark*

7. Outkast3r09

i'm pretty sure i have it down so you can come check it after $y''-xy=\sum_{n=2}^\infty c_nn(n-1)x^{n-2}-x\sum_{n=0}^\infty c_nx^n$

8. Outkast3r09

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

9. Outkast3r09

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

10. Outkast3r09

$\sum_{k=0}^\infty c_{k+2}(k+2)(k+1)x^k-\sum_{k=1}^\infty c_{k-1}x^k$ where $k=n+1$ $k-1=n$

11. Outkast3r09

pull out one term k=0 of first $2c_2+\sum_{k=1}^\infty c_{k+2}(k+2)(k+1)x^k-\sum_{k=1}^\infty c_{k-1}x^k$

12. Outkast3r09

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

13. Outkast3r09

since identity 0=0 the sum and $2c_2=0$ $c_2=0$

14. Outkast3r09

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

15. Outkast3r09

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

16. Outkast3r09

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

17. Outkast3r09

k=5 $c_7=\frac{c_4}{7*6}=\frac{c_1}{7*6*4*3}$

18. Outkast3r09

$c_8=c_5=0$

19. Outkast3r09

@TuringTest look correct so far?

20. TuringTest

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

21. experimentX

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

22. TuringTest

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

23. Outkast3r09

Is zarkon the computer on lol

24. experimentX

these standard equations power solutions look intimidating ...

25. Outkast3r09

$y=c_0+c_1x+0+\frac{c_0}{3*2}x^3+\frac{c_1}{4*3}x^4+0+\frac{c_0}{2*3*5*6}x^6$ $+\frac{c_1}{3*4*6*7}x^7+0......$

26. Outkast3r09

$y_1(x)=1+\frac{1}{2*3}x^3+\frac{1}{2*3*5*6}x^6...$ $y_2(x)=x+\frac{1}{3*4}x^4+\frac{1}{3*4*6*7}x^7.......$

27. experimentX

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

28. experimentX

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

29. Outkast3r09

thats what i have

30. Outkast3r09

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

31. experimentX

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

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