- #1
Ted123
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Homework Statement
[PLAIN]http://img59.imageshack.us/img59/2091/diffeq.png
[PLAIN]http://img684.imageshack.us/img684/6748/diffeqp.png
The Attempt at a Solution
Making the substitutions [tex]y= \sum_{n=0}^{\infty} a_n x^n[/tex] and [tex]y^{\prime} = \sum_{n=0}^{\infty}na_nx^{n-1},[/tex]
[tex]\begin{align*}
y'-2xy & = \sum_{n=0}^{\infty} (na_nx^{n-1} - 2xa_nx^n )\\
&= \sum_{n=0}^{\infty} na_nx^{n-1} - \sum_{n=0}^{\infty} 2xa_nx^n\\
&= \sum_{n=1}^{\infty} na_nx^{n-1} - 2\sum_{n=0}^{\infty} a_nx^{n+1}\\
&= \sum_{n=1}^{\infty} na_nx^{n-1} - 2\sum_{n=0}^{\infty} a_nx^{n+1}
\end{align*}
[/tex]
Now I'm having trouble seeing how to shift the summation index to combine the 2 sums into one and to have one coefficient of [tex]x^{n+1}[/tex]
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