## anonymous 5 years ago Use power series to solve y''(t)=y*y'(t), y(0)=2 and y'(0)=-3 I know to assume that y=a0+a1t+a2t^2+a3t^3+a4t^4 (where the number following a is its subscript and not a coefficient on t) and I know to get the first and second derivatives from there. What I am having trouble with is foiling the series for y and the series for y'. Could someone please help me finish this problem? If I could just multiply the two series together without any mistakes, I'd pretty much be done.

1. anonymous

2. anonymous

thank you so much for that

3. anonymous

So it's okay?

4. anonymous

Would you be willing to help me foil out the two series? My teacher wants me to show them out to the fourth degree (something I forgot to write into my original question)

5. anonymous

the series for y and y' that is

6. anonymous

Do you mean your teacher wants you to expand to the fourth degree?

7. anonymous

yeah. I actually just did the foil for the fourth degree of y and y' . y*y'=a0a1+(2a0a2+(a1)^2)t+(3a0a3+3a1a2)t^2+(4a0a4+4a1a3+2(a2^2)t^3+(5a0a5+5a1a4+5a2a3)t^4. Here's my new problem though: Now that I have all of the values for them, I am told to equate different coefficients of powers of t, but I keep on screwing up the scratchwork. would you help me euate the different powers of t?

8. Gina

$y \prime=10^{x+y}$

9. anonymous

10. anonymous

That looks right. I haven't completed a total expansion, though. So, you're now needing to actually determine the coefficients? Are you allowed to use the formula I derived earlier?

11. anonymous

$a_{n+2}=\frac{1}{(n+1)(n+2)}\sum_{k=0}^{n}(n-k+1)a_na_{n-k+1}$

12. anonymous

Yeah, but I think he just wants us to equate the different coefficients of different powers of t and solve algebraically. So for instance, the constant term for y''=yy' is 2a2=a0a1-->a2=(1/2)a0a1. then you go down the list and just put everything in terms of a0 and a1.

13. anonymous

Gina, post your question again separately and I'll help.

14. anonymous

does that make sense?

15. anonymous

Yep. You have a0 and a1...the rest is just turgid algebra. I get the feeling you're screwing it up because it's tedious and has many 'moving parts'. It sounds like you know what to do, though. I don't know what else I can do.

16. anonymous

Yeah, it sounds like your teacher just wants you to go through the motions.

17. anonymous

help me with the actual algebra part? I really need to get this question done and all that I'm screwing up is the tedious part.

18. anonymous

I'll see if I can get you started.

19. anonymous

thank you so much. I think I can get the first few t terms without trouble. It's like the t^3 and t^4 terms taht are giving me trouble.

20. anonymous

any luck?

21. anonymous

I expanded first to check, which took some time.

22. anonymous

$2a_2=a_0a_1\rightarrow a_2=\frac{2 \times -3}{2}=-3$ $6a_3=2a_0a_2+a^2_1\rightarrow a_3=\frac{2 \times 2 \times -3+(-3)^2}{6}=-\frac{1}{2}$ Is this looking familiar?

23. anonymous

$12a_4=3a_0a_3+3a_1a_2$$\rightarrow a_4=\frac{3*2*-\frac{1}{2}+3*-3*-3*-3}{12}=-7$

24. anonymous

$20a_5=4a_0a_4+4a_1a_3+2a^2_2\rightarrow a_5=-\frac{8}{5}$

25. anonymous

$30a_6=5a_0a_5+5a_1a_4+5a_2a_3 \rightarrow a_6=\frac{193}{60}$