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anonymous
 5 years ago
PLEASE HELP!!! Use power series to solve y''(t)=y*y'(t), y(0)=2 and y'(0)=3
anonymous
 5 years ago
PLEASE HELP!!! Use power series to solve y''(t)=y*y'(t), y(0)=2 and y'(0)=3

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0You start with a series expansion for y\[y=\sum_{n=0}^{\infty}a_nt^n\] and take the necessary derivatives,\[y'=\sum_{n=0}^{\infty}na_nt^{n1}\]and\[y''=\sum_{n=0}^{\infty}n(n1)a_nt^{n2}\]Since the first term of y' and first two terms of y'' equal zero, you have,\[y'=\sum_{n=1}^{\infty}na_nt^{n1}\]and\[y''=\sum_{n=2}^{\infty}n(n1)a_nt^{n2}\]Now, because we want to add and multiply series and derive information on the coefficients, we need to ensure each sum expresses everything in terms of t^n. Hence, we transform n to n+1 in y' and n to n+2 in y''. If you do this, and expand, you'll see you haven't actually changed the sum, just the form. Doing this, and using your equation of y''=y*y', we have\[\sum_{n=0}^{\infty}(n+1)(n+2)a_{n+2}t^n=\sum_{n=0}^{\infty}a_nt^n \times \sum_{n=0}^{\infty}(n+1)a_{n+1}t^n\]The last product is a Cauchy product (http://\[(n+1)(n+2)a_{n+2}=\sum_{k=0}^{n}(nk+1)a_na_{nk+1}\]en.wikipedia.org/wiki/Cauchy_product) which allows us to express the righthand side as\[\sum_{n=0}^{\infty}\left( \sum_{k=0}^{n}(nk+1)a_na_{nk+1} \right)t^n\]Since the lefthand side and righthand side are equal if and only if for each t^n the coefficients are equal,\[(n+1)(n+2)a_{n+2}=\sum_{k=0}^{n}(nk+1)a_na_{nk+1}\]That is,\[a_{n+2}=\frac{1}{(n+1)(n+2)}\sum_{k=0}^{n}(nk+1)a_na_{nk+1}\]

Gina
 5 years ago
Best ResponseYou've already chosen the best response.0o mr.lokisan i have a maths proces error here so i can not see the information

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0You use your initial conditions with the definition for y above to determine the first two coefficients you'll need in order to use that formula above to generate the rest.\[y(0)=\sum_{n=0}^{\infty}a_nt^n=a_0+a_1(0)+a_2(0)^2+...=a_0\]Since \[y(0)=2, a_0=2\]Similarly,\[y'(0)=\sum_{n=0}^{\infty}(n+1)a_{n+1}t^n=a_1+2a_2(0)+3a_3(0)^2+...=a_1\]So\[y'(0)=3, a_1=3\]

Gina
 5 years ago
Best ResponseYou've already chosen the best response.0o right but your information is writing maths error which means the symboys u used here ,,,so i can not see any formula plaese help by typing them here on my mail ;geo5phiri@yahoo.com

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I thought this question was for omnideus. You need it too? I'll have to take screen shots and mail them. Some of your own symbols aren't coming through. Can I confirm your mail is geo5phiri@yahoo.com?

Gina
 5 years ago
Best ResponseYou've already chosen the best response.0yes it is ,,,that is my mail

Gina
 5 years ago
Best ResponseYou've already chosen the best response.0wow)) mr.lokisan am yo number fan,,could just teach me differentuation in general?? but i some basics

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Hehe. Are my answers coming through for the other questions? Can you see the symbols?
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