# DE: Lower Bound for radius of convergence

1. Jul 22, 2012

### BobMarly

Prb:(x^4+4*x^2+16)y"+4(x-1)y'+6xy=0
P=(x^4+4*x^2+16) Q=4(x-1) R=6x
P=0 for - 1 - 3^(1/2)*i
1 - 3^(1/2)*i
- 1 + 3^(1/2)*i
1 + 3^(1/2)*i
Q=0 for 1
R=0 for 0
Do we ignore Q & R, plotting P, then find shortest distance which would equal 2?

2. Jul 22, 2012

### HallsofIvy

Staff Emeritus
Basically, yes. If P were never 0, all solutions to the differential equation would be analytic. Because P is 0 at those four complex numbers, we can take the solution to be 0 away from them. The distance from 0 to those four points is 2 so the radius of convergence of a power series for the solution, centered at 0 wil have radius of convergence 2.