Solution Series Near An Ordinary Point

[jjm5119]

Using the soltion series n=0 (summation) to infinity of a(sub n)*x^n about x(sub 0)=0 of the ODE (1+x^2)y''+y'+2y=0
ives the following relations between the coefficients a(sub n)

a2= _____a1 + _____a0
a3= _____a1 + _____a0
a(n+2)= _____a(n+1)+______a(sub n) for n >= 2

so first i divide by (1+x^2) to make the coefficient of y'' 1 to get y''+(y/(1+x^2))+2y/(1+x^2)

i then find the first and second derivatives of these power series to get
y' equal to --> n=0 (summation) to infinity of (n+1)a(sub n+1) * x^n
y'' equal to --> n=0 (summation) to infinity of (n+2)(n+1)a(sub n+2) * x^n

now i have no idea what to do. any help is appreciated.

 

[Deleted member 4993]

Now put those back into the original ODE and gather coefficients of \(\displaystyle x^i\) together and set those to zero.

 

[jjm5119]

ok i solved it thanks