- #1
Pinedas42
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Homework Statement
Find the indicial equation and find 2 independent series solutions for the DE:
xy''-xy'-y=0 about the regular singular point x=0
Homework Equations
y=Ʃ(0→∞) Cnxn+r
y'=Ʃ(0→∞) Cn(n+r)xn+r-1
y''=Ʃ(0→∞) Cn(n+r)(n+r-1)xn+r-2
The Attempt at a Solution
Finding the indicial eq.
Stan. form y''-y'-(1/x)y=0
p(x)=x*(-1)=-x
q(x)=x2*(-1/x)=-x
Making ao and bo both zero for
r(r-1)+aor+bo=0
so r=0,1
Solving for the equation I finish with (I'll skip a few steps, confident in this portion)
Cor(r-1)xr-1+Ʃ(0→∞) [Cn+1(n+r+1)(n+r)-Cn(n+r+1)]xn+r
Inside the brackets = 0 so the recurrence relation is
Cn=Cn+1(n+r) , n=0,1,2,3...
For r=1, Cn=Cn+1(n+1)
Co=C1, n=0
C1=C2(2)=Co/2, n=1
C2=C3(3)=Co/2*3, n=2
C3=C4(4)=Co/2*3*4, n=3
I conclude y1=Co(1+x+x2/2!+x3/3!...)
which is the series for ex, though our professor wants this in series form.
For r=0, Cn=Cn+1(n)
Co=0, n=0
C1=C2, n=1
C2=C3(2)=C1/2, n=2
C3=C4(3)=C1/2*3, n=3
I'm not to sure how put this into summing terms, the zero is throwing me off.
I'd like to know if I'm on the right track with this. I feel like I did everything as I was supposed to, but something is giving me gut feeling that some portion is erroneous.
Thanks for the help, if you choose to lend it to this tedious problem lol :zzz: