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**Belowzero78** Question: The differential equation $\displaystyle y'' + ty' + (t^2-2)y = 0, y(0) = -2, y'(0) = -2 $ has a power series solution about t = 0 of the form $\displaystyle $\displaystyle\sum\limits_{n=0}^\infty c_{n}t^{n}$ $. The first six coefficients are: $\displaystyle C_{0}, C_{1}, C_{2}, C_{3}, C_{4}, C_{5} = ? $

I applied the shift when trying to make the all the summations having $\displaystyle t^{n} $, but i get stuck at this step:

$\displaystyle $\displaystyle\sum\limits_{n=0}^\infty c_{n+2}(n+2)(n+1)t^{n}$ + $\displaystyle\sum\limits_{n=0}^\infty c_{n+1}(n+1)t^{n}$ + $\displaystyle\sum\limits_{n=0}^\infty c_{n}t^{n+2}$ + $\displaystyle\sum\limits_{n=0}^\infty 2c_{n}t^{n}$ $

Like how do i get rid of the $\displaystyle t^{n+2} $ in the third summation there?