find the series solution,power series


by dp182
Tags: around a point, ode, power series, series solution
dp182
dp182 is offline
#1
Jul3-11, 10:52 PM
P: 22
1. The problem statement, all variables and given/known data
2xy''-x(x-1)y'-y=0 about x=0
what are the roots of the indicial equation and for the roots find the recurrence relation that defines the the coef an

2. Relevant equations
2xy''-x(x-1)y'-y=0 about x=0

assuming the solution has the form y=[itex]\Sigma[/itex]anxn+r
y'=[itex]\Sigma[/itex](n+r)anxn+r-1
y''=[itex]\Sigma[/itex](n+r)(n+r-1)anxn+r-2

3. The attempt at a solution
after plugging into the solution I get
2[itex]\Sigma[/itex](n+r)(n+r-1)anxn+r-1-[itex]\Sigma[/itex](n+r)anxn+r+1-[itex]\Sigma[/itex](n+r)anxn+r-1-[itex]\Sigma[/itex]anxn+r
then I attempt to make all the x's the same and and make the sigma's equal so after doing that I get
2[itex]\Sigma[/itex](n+r+1)(n+r)an+1xn+r-[itex]\Sigma[/itex](n+r-1)an-1xn+r-[itex]\Sigma[/itex](n+r+1)an+1xn+r-[itex]\Sigma[/itex]anxn+r
I know that I need to replace the 0 under the sigma's with a (-1) on terms 1,3 but term 2 is whats throwing me off any help would be great
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
vela
vela is offline
#2
Jul4-11, 02:17 AM
Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,530
The third sum should be positive. Explicitly writing in the limits, you have
[tex]2\sum_{n=0}^\infty (n+r)(n+r-1)a_n x^{n+r-1}
-\sum_{n=0}^\infty (n+r) a_n x^{n+r+1}
+\sum_{n=0}^\infty (n+r)a_n x^{n+r-1}
-\sum_{n=0}^\infty a_n x^{n+r} = 0[/tex]
As you noted, only the first and third sums give you a [itex]x^{r-1}[/itex] term, and the second sum doesn't give you an [itex]x^r[/itex] term. Separating those terms out, you have
\begin{eqnarray*}
&&[2r(r-1) + r]a_0x^{r-1} + \\
&&[2(r+1)r a_1 + (r+1)a_1 - a_0]x^r + \\
&&2\sum_{n=1}^\infty (n+r+1)(n+r)a_{n+1}x^{n+r}
-\sum_{n=1}^\infty (n+r-1)a_{n-1}x^{n+r}
+\sum_{n=1}^\infty (n+r+1)a_{n+1}x^{n+r}
-\sum_{n=1}^\infty a_n x^{n+r} = 0
\end{eqnarray*}
By assumption, a0 isn't equal to 0, so you must have 2r(r-1)+r=0. That's your indicial equation.


Register to reply

Related Discussions
series solution up to a term, power series Calculus & Beyond Homework 5
Find Sum of Series Using Power Series Calculus & Beyond Homework 15
Power series solution to DE Calculus & Beyond Homework 10
Find the closed form of power series, that is solution to the given diff. equation Calculus & Beyond Homework 4
Exploiting Geometric Series with Power Series for Taylors Series Calculus & Beyond Homework 11