- #1
jakobsandberg
- 4
- 0
Homework Statement
Find the indicial roots of the following Differential Equation: xy'' - y' + x3y = 0
Homework Equations
y = Ʃ[n=0 to infinity]cnxn+r
y' = Ʃ[n=0 to infinity](n+r)cnxn+r-1
y'' = Ʃ[n=0 to infinity](r+r)(n+r-1)cnxn+r-2
The Attempt at a Solution
Plugging these values into the differential equation, I got
xr{Ʃ[n=0 to infinity](n+r)(n+r-1)cnxn-1 - Ʃ[n=0 to infinity](n+r)cnxn-1 + Ʃ[n=0 to infinity]4cnxn+3} = 0
The three sums must produce the x to the same exponent, so I tried pulling out the first 4 terms of the first two sums, so the three sums would each output x3 as their first term [the first two sums starting from n=4]. However, this left me with the following equation:
r(r-1)c0x-1 - rc0x-1 + r(r+1)c1 - (r+1)c1 + (r+1)(r+2)c2x - (r+2)c2x - (r+2)(r+3)c3x2 - (r+3)c3x2 + [remaining sums] = 0.
How do I solve for r with this equation? I don't know how to find the roots.
[the solution to the DE is y=c1cos(x2) + c2sin(x2)
Last edited: