Power Series Solution to Linear ODE

1. The problem statement, all variables and given/known data

Let y(x)=[itex]\sum[/itex]c_kx^k (k=0 to ∞) be a power series solution of



(x²-1)y''+x³y'+y=2x, y(0)=1, y'(0)=0



Note that x=0 is an ordinary point.



2. Relevant equations

y(x)=[itex]\sum[/itex]c_kx^k (k=0 to ∞)

y'(x)=[itex]\sum[/itex](kc_kx^k-1) (k=1 to ∞)

y''(x)=[itex]\sum[/itex](k(k-1))c_kx^k-2 (k=2 to ∞)



3. The attempt at a solution



(x²-1)[itex]\sum[/itex](k(k-1))c_kx^k-2 (k=2 to ∞) +[itex]\sum[/itex](kc_kx^k) (k=1 to ∞)+[itex]\sum[/itex]c_kx^k (k=0 to ∞) -2x=0



Making all the series start with the x³ term

2C₂x⁰+[itex]\sum[/itex](k(k-1))c_kx^k (k=3 to ∞)- 2C₂x⁰-6C₃x¹-12C₄x²-[itex]\sum[/itex](k(k-1))c_kx^k-2 (k=5 to ∞) + [itex]\sum[/itex](kc_kx^k+2) (k=1 to ∞)+C₀x⁰+C₁x¹+C₂x²+[itex]\sum[/itex]c_kx^k (k=3 to ∞)





2C₂x⁰+[itex]\sum[/itex](n(n-1))c_nxⁿ (n=3 to ∞)- 2C₂x⁰-6C₃x¹-12C₄x²-[itex]\sum[/itex](n+2)(n+1))c_n+2xⁿ (n=3 to ∞) + [itex]\sum[/itex]((n-2)c_n-2xⁿ) (n=3 to ∞)+C₀x⁰+C₁x¹+C₂x²+[itex]\sum[/itex]c_nxⁿ (n=3 to ∞)







2C₂x⁰-2C₂x⁰-6C₃x¹-12C₄x²+C₀x⁰+C₁x¹+C₂x²=0

c₀=0

c₃={c₁-2}/{6}

c₄={c₂}/{12}



c_n+2={n(n-1)c_n+(n-2)c_n-2+c_n}/{(n+2)(n-1)} n=3,4,5...





I started with c₅={(7c₃+c₁)}/{10}

but both of my c₁ and c₃ terms are tied up in the same equation, not really sure were to go from here

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