I have the following differential equation:
$(1-t^2)\frac{d^2x}{dt^2}-2t\frac{dx}{dt}+6x=0$
And I have the initial conditions as follows:
$x(0)=1, \frac{dx}{dt}=0$
I know power series can be used to solve this ODE, and also know the answer is $x = 1-3t^2$.
Any kind of help will be appreciated, thanks in advance!
Your ODE is Legendre's equation $$ (1-t^2)\frac{d^2x}{dt^2}-2t\frac{dx}{dt}+\ell (\ell + 1)x=0 $$ with $\ell = 2$ so the solution to that is the Legendre polynomial of degree 2 $$ P_2(t)=\frac{1}{2}(3t^2-1) $$ This is still a solution when multiplied by a constant. If you multiply by $-2$ it satisfies your initial conditions.