5. Find at least the first 4 nonzero terms of the solution to (2 + 4x-2x^2)y"-12(z-1)y'-12y = 0, y(1) = 0, y'(1) =-1.

We're set up to find a series solution to the ODE centered around [tex]x=1[/tex]. Let's first replace [tex]z=x-1[/tex], so that[tex]y(x)=y(z+1)\implies y'(x)=y'(z+1)\implies y''(x)=y''(z+1)[/tex]and we'll look for a series solution centered at [tex]z=0[/tex]. We have[tex]y=\displaystyle\sum_{n\ge0}a_nz^n[/tex][tex]\implies y'=\displaystyle\sum_{n\ge0}(n+1)a_{n+1}z^n[/tex][tex]\implies y''=\displaystyle\sum_{n\ge0}(n+2)(n+1)a_{n+2}z^n[/tex]The ODE is then[tex](2+4x-2x^2)y''-12(x-1)y'-12y=\left(4-2(x-1)^2\right)y''-12(x-1)y'-12y=0[/tex]which in terms of [tex]z[/tex] is[tex]\left(4-2z^2\right)y''-12zy'-12y=0[/tex]Substituting the series above into the ODE gives[tex]\displaystyle4\sum_{n\ge0}(n+2)(n+1)a_{n+2}z^n-2\sum_{n\ge0}(n+2)(n+1)a_{n+2}z^{n+2}-12\sum_{n\ge0}(n+1)a_{n+1}z^{n+1}-12\sum_{n\ge0}a_nz^n=0[/tex][tex]\displaystyle4\sum_{n\ge0}(n+2)(n+1)a_{n+2}z^n-2\sum_{n\ge2}n(n-1)a_nz^n-12\sum_{n\ge1}na_nz^n-12\sum_{n\ge0}a_nz^n=0[/tex][tex]\displaystyle\left(8a_2+24a_3z+4\sum_{n\ge2}(n+2)(n+1)a_{n+2}z^n\right)-2\sum_{n\ge2}n(n-1)a_nz^n-12\left(a_1z+\sum_{n\ge2}na_nz^n\right)-12\left(a_0+a_1z+\sum_{n\ge2}a_nz^n\right)=0[/tex][tex]\displaystyle(8a_2-12a_0)+(24a_3-24a_1)z+\sum_{n\ge2}\bigg[(4n^2+3n+2)a_{n+2}-(2n^2+10n+2)a_n\bigg]z^n=0[/tex]so that the coefficients follow the recurrence,[tex]\begin{cases}a_0=y(0)=0\\\\a_1=y'(0)=-1\\\\a_n=\dfrac{2(n-2)^2+10(n-2)+2}{4(n-2)^2+3(n-2)+2}a_{n-2}=\dfrac{2n^2+2n-10}{4n^2-13n+12}a_{n-2}&\text{for }n\ge2\end{cases}[/tex]Thanks to the dependency between every other coefficient, we have [tex]a_n=0[/tex] for all even-valued [tex]n[/tex]. Meanwhile,[tex]a_1=-1[/tex][tex]a_3=-\dfrac{14}9[/tex][tex]a_5=-\dfrac{700}{423}[/tex][tex]a_7=-\dfrac{23,800}{16,497}[/tex]so that[tex]y(z)\approx-z-\dfrac{14}9z^3-\dfrac{700}{423}z^5-\dfrac{23,800}{16,497}z^7[/tex]or in terms of [tex]x[/tex],[tex]\boxed{y(x)\approx-(x-1)-\dfrac{14}9(x-1)^3-\dfrac{700}{423}(x-1)^5-\dfrac{23,800}{16,497}(x-1)^7}[/tex]