Changes

<math>(2a_2)+(6a_3-a_1)x+(12a_4-2a_2+a_0)x^2 + \dots </math>

<math>(2a_2)+(6a_3-a_1)x+(12a_4-2a_2+a_0)x^2 + \dots </math>

so that for <math>n\geq 2</math>, the coefficient of <math>x^n</math> on the right hand side is  <math>(n+1)(n+2)a_{n+2}-na_n+a_{n-2}</math>.  So we need only set this equal to <math>1/(n!)</math> to find an infinite series representation of our solution.

so that for <math>n\geq 2</math>, the coefficient of <math>x^n</math> on the right hand side is  <math>(n+1)(n+2)a_{n+2}-na_n+a_{n-2}</math>.  So we need only set this equal to <math>1/(n!)</math> to find an infinite series representation of our solution.  If we're given the values of <math>y(0), y'(0)</math>, as we frequently are in applications (remember what we said about having two "degrees of freedom" in a DEQ involving second-order derivatives?) we can solve this completely.  Let's say <math>y(0)=0, y'(0)=1</math>.  We have

 

 

This method may not seem particularly useful.  After all, unless we can find a closed form solution for the <math>a_n</math>, we are going to have to keeping solving this linear equations in <math>a_n</math> forever to get the solution.  However, it may happen, on occasion, that we ''can'' find such a closed form representation.  Or, it may happen that we find no closed form, but recognize it as similar or equivalent to a function which has been studied for decades or centuries, so that all the properties we wish to know about are easily available to us.  Or, we may not ''care'' about a closed form - near 0, those higher powers will become very small, and a low-order approximation becomes highly accurate.

===Functional Transformations===

===Functional Transformations===