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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.
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.

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I answered your question earlier.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0thank you so much for that

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Would 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)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0the series for y and y' that is

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Do you mean your teacher wants you to expand to the fourth degree?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0yeah. 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?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I'm not sure how your reply makes sense here Gina.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0That 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?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[a_{n+2}=\frac{1}{(n+1)(n+2)}\sum_{k=0}^{n}(nk+1)a_na_{nk+1}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Yeah, 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Gina, post your question again separately and I'll help.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0does that make sense?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Yep. 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Yeah, it sounds like your teacher just wants you to go through the motions.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0help 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I'll see if I can get you started.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0thank 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.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I expanded first to check, which took some time.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[2a_2=a_0a_1\rightarrow a_2=\frac{2 \times 3}{2}=3\] \[6a_3=2a_0a_2+a^2_1\rightarrow a_3=\frac{2 \times 2 \times 3+(3)^2}{6}=\frac{1}{2}\] Is this looking familiar?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[12a_4=3a_0a_3+3a_1a_2\]\[\rightarrow a_4=\frac{3*2*\frac{1}{2}+3*3*3*3}{12}=7\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[20a_5=4a_0a_4+4a_1a_3+2a^2_2\rightarrow a_5=\frac{8}{5}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[30a_6=5a_0a_5+5a_1a_4+5a_2a_3 \rightarrow a_6=\frac{193}{60}\]
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