Power Series & Singular Points: Why Change the Form?

[CPL.Luke]

when finding a power series solution we have to put the differential equation

ay''+by'+c=0

into the form

y''+By+C=0

this leads to singular points when a=0 but why can't we leave the equation in its original form and use power series substitution to avoid singular points?

or in other words why do we care about the second form?

 

[Hurkyl]

Experiment. Try it out and see. Take a simple example,

x y'' - 1 = 0

for example, and see what happens.