Question

Find the power series in x-xo for the general solution of y"-y=0; xo=3.

Answer 1

@satellite73 @Directrix @Nnesha @Astrophysics

Answer 2

You're assuming a solution exists of the form \(y=\sum\limits_{n\ge0}a_n (x-x_0)^n\). You can make the derivation of \(y\) a bit simpler with a substitution that just shifts the argument of the series. Let \(z=x-x_0\), then the solution is of the form \(y=\sum\limits_{n\ge0}a_nz^n\). Do you know how to work from there?

Answer 3

With \(y=\sum\limits_{n\ge0}a_nz^n\), you have
\[\sum_{n\ge2}n(n-1)a_nz^{n-2}-\sum_{n\ge0}a_nz^n=0\]Shifting the index of the first sum, you can rewrite this as
\[\sum_{n\ge0}(n+2)(n+1)a_{n+2}z^n-\sum_{n\ge0}a_nz^n=0\]then combine the terms to get
\[\sum_{n\ge0}\bigg[(n+2)(n+1)a_{n+2}-a_n\bigg]z^n=0\]Solve the recurrence, find the solution, then back-substitute the \(z\) for \(x-3\).