Question

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

Answer 1

You start with a series expansion for y\[y=\sum_{n=0}^{\infty}a_nt^n\] and take the necessary derivatives,\[y'=\sum_{n=0}^{\infty}na_nt^{n-1}\]and\[y''=\sum_{n=0}^{\infty}n(n-1)a_nt^{n-2}\]Since the first term of y' and first two terms of y'' equal zero, you have,\[y'=\sum_{n=1}^{\infty}na_nt^{n-1}\]and\[y''=\sum_{n=2}^{\infty}n(n-1)a_nt^{n-2}\]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://\[(n+1)(n+2)a_{n+2}=\sum_{k=0}^{n}(n-k+1)a_na_{n-k+1}\]en.wikipedia.org/wiki/Cauchy_product) which allows us to express the right-hand side as\[\sum_{n=0}^{\infty}\left( \sum_{k=0}^{n}(n-k+1)a_na_{n-k+1} \right)t^n\]Since the left-hand side and right-hand side are equal if and only if for each t^n the coefficients are equal,\[(n+1)(n+2)a_{n+2}=\sum_{k=0}^{n}(n-k+1)a_na_{n-k+1}\]That is,\[a_{n+2}=\frac{1}{(n+1)(n+2)}\sum_{k=0}^{n}(n-k+1)a_na_{n-k+1}\]

Answer 2

o mr.lokisan i have a maths proces error here so i can not see the information

Answer 3

You 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\[y'(0)=-3, a_1=-3\]

Answer 4

o 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

Answer 5

I 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?

Answer 6

yes it is ,,,that is my mail

Answer 7

wow)) mr.lokisan am yo number fan,,could just teach me differentuation in general?? but i some basics

Answer 8

Hehe. Are my answers coming through for the other questions? Can you see the symbols?