Find a general solution using power swries: ((x^2)+1)y'' + 6xy' + 6y =0

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If `y= sum_(n=0)^(oo)a_nx^n` is solution to the equation `(x^2+1)y'' + 6xy' + 6y =0`

 

 

`y'=sum_(n=1)^(oo)na_nx^(n-1)`

 

`y''=sum_(n=2)^(oo)(n(n-1)a_nx^(n-2)`

 

`(1+x^2)y''=sum_(n=2)^(oo)n(n-1)a_nx^(n-2)+sum_(n=2)^(oo)n(n-1)a_nx^(n-2)x^2`

`(1+x^2)y''=sum_(n=2)^(oo)n(n-1)a_nx^(n-2)+sum_(n=2)^(oo)n(n-1)a_nx^n`

 

`xy'=sum_(n=1)^(oo)na_nx^(n-1)x=sum_(n=1)^(oo)na_nx^(n)`

 

If we plug it in the differential equation,

`sum_(n=2)^(oo)n(n-1)a_nx^(n-2)+sum_(n=2)^(oo)n(n-1)a_nx^n+6sum_(n=1)^(oo)na_nx^(n)+ 6sum_(n=0)^(oo)a_nx^n=0`

In the first sum, make the...

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