SigepBrandon
New member
- Joined
- Feb 17, 2011
- Messages
- 39
The complete problem is as follows:
Use power series to find the solution to the initial value problem y''+3xy'-6y=1, y(0)=a0, y'(0)=0 by letting \(\displaystyle \sum_{k=0}^{\infty} a_{k}x^{k}=1\) and finding a recurrence relation for the coefficients ak.
I found the derivatives of the series to be:
\(\displaystyle y'=\sum_{k=1}^{\infty} ka_{k}x^{k-1}\) and \(\displaystyle y''=\sum_{k=2}^{\infty} k(k-1)a_{k}x^{k-2}\)
plugging the derivatives of the series into the differential equation:
\(\displaystyle \sum_{k=2}^{\infty} k(k-1)a_{k}x^{k-2}+3x \sum_{k=1}^{\infty} ka_{k}x^{k-1} - 6 \sum_{k=0}^{\infty} a_{k}x^{k}=1\)
I know I need to multiply the coefficients (3x and 6) into their respective series and then perform the shift in order to combine. However I do not know how to get the coefficients into the same form. Any help would be greatly appreciated.
-Brandon
Use power series to find the solution to the initial value problem y''+3xy'-6y=1, y(0)=a0, y'(0)=0 by letting \(\displaystyle \sum_{k=0}^{\infty} a_{k}x^{k}=1\) and finding a recurrence relation for the coefficients ak.
I found the derivatives of the series to be:
\(\displaystyle y'=\sum_{k=1}^{\infty} ka_{k}x^{k-1}\) and \(\displaystyle y''=\sum_{k=2}^{\infty} k(k-1)a_{k}x^{k-2}\)
plugging the derivatives of the series into the differential equation:
\(\displaystyle \sum_{k=2}^{\infty} k(k-1)a_{k}x^{k-2}+3x \sum_{k=1}^{\infty} ka_{k}x^{k-1} - 6 \sum_{k=0}^{\infty} a_{k}x^{k}=1\)
I know I need to multiply the coefficients (3x and 6) into their respective series and then perform the shift in order to combine. However I do not know how to get the coefficients into the same form. Any help would be greatly appreciated.
-Brandon