Question

how do you know when to use power series and when to use euler's equation?

Answer 1

@manjuthottam , Can you give an example because your question seems vague. If the question is in context of example, then maybe I can help you

Answer 2

here's an attachment of euler's method

Answer 3

here's an attachment of power series

Answer 4

this is for differential equations class

Answer 5

@manjuthottam , ok. I get it now. Sorry, I haven't learnt this topic yet. But as far as my observation's go. In the euler's method, the intiial value is given i.e y(1) , y'(1)

But in the case of Power series problem.the initial values are not given.

Answer 6

oh ok! thank you :)

Answer 7

BTW @ manjuhottam, IVP-->Intial value problem :D

Answer 8

ok :)

Answer 9

I think this has to do with the following: In the Euler formula, we are looking for a solution to the equation:\[ax^2y'' + bxy' + cy = 0\]around \(x_0 = 0 \). In the power series solution, we want to find the solution to: \[ p(x)y'' + q(x)y' + r(x)y = 0\]around some \(x_0\) which is an ordinary point. An ordinary point is a point \(x_0 \) such that: \[\frac{q(x)}{p(x)} and \frac{r(x)}{p(x)}\]are analytic, that is, both have a Taylor series of the form: \[\sum_{n=0}^{\infty} a_n(x-x_0)^n \]For some ordinary point \( x_0\).

Answer 10

I should've been clearer: the solution has that Taylor series around an ordinary point. We only need to ensure that q/p and r/p have a Taylor Series on x = x0.