[tex]y=\displaystyle\sum_{n\ge0}a_nx^n=a_0+\sum_{n\ge1}a_nx^n[/tex]
[tex]y'=\displaystyle\sum_{n\ge0}(n+1)a_{n+1}x^n=a_1+\sum_{n\ge0}(n+1)a_{n+1}x^n[/tex]
[tex]y''=\displaystyle\sum_{n\ge0}(n+1)(n+2)a_{n+2}x^n[/tex]
Notice that [tex]y(0)=-3=a_0[/tex], and [tex]y'(0)=2=a_1[/tex].
Substitute these series into the ODE:
[tex](x-1)y''-xy'+y=0[/tex]
[tex]\displaystyle\sum_{n\ge0}(n+1)(n+2)a_{n+2}x^{n+1}-\sum_{n\ge0}(n+1)(n+2)a_{n+2}x^n-\sum_{n\ge0}(n+1)a_{n+1}x^{n+1}+\sum_{n\ge0}a_nx^n=0[/tex]
Shift the indices to get each series to include a [tex]x^n[/tex] term.
[tex]\displaystyle\sum_{n\ge1}n(n+1)a_{n+1}x^n-\sum_{n\ge0}(n+1)(n+2)a_{n+2}x^n-\sum_{n\ge1}na_nx^n+\sum_{n\ge0}a_nx^n=0[/tex]
Remove the first term from both series starting at [tex]n=0[/tex] to get all the series starting on the same index [tex]n=1[/tex]:
[tex]\displaystyle-2a_2+a_0+\sum_{n\ge1}n(n+1)a_{n+1}x^n-\sum_{n\ge1}(n+1)(n+2)a_{n+2}x^n-\sum_{n\ge1}na_nx^n+\sum_{n\ge1}a_nx^n=0[/tex]
[tex]\displaystyle-2a_2+a_0+\sum_{n\ge1}\bigg[n(n+1)a_{n+1}-(n+1)(n+2)a_{n+2}-(n-1)a_n\bigg]x^n=0[/tex]
The coefficients are given recursively by
[tex]\begin{cases}a_0=-3\\a_1=2\\\\a_n=\dfrac{(n-2)(n-1)a_{n-1}-(n-3)a_{n-2}}{n(n-1)}&\text{for }n>1\end{cases}[/tex]
Let's see if we can find a pattern to these coefficients.
[tex]a_2=\dfrac{a_0}2=-\dfrac32=-\dfrac3{2!}[/tex]
[tex]a_3=\dfrac{2a_2}{3\cdot2}=-\dfrac12=-\dfrac3{3!}[/tex]
[tex]a_4=\dfrac{2\cdot3a_3-a_2}{4\cdot3}=-\dfrac18=-\dfrac3{4!}[/tex]
[tex]a_5=\dfrac{3\cdot4a_4-2a_3}{5\cdot4}=-\dfrac1{40}=-\dfrac3{5!}[/tex]
[tex]a_6=\dfrac{4\cdot5a_5-3a_4}{6\cdot5}=-\dfrac1{240}=-\dfrac3{6!}[/tex]
and so on, suggesting that
[tex]a_n=-\dfrac3{n!}[/tex]
which is also consistent with [tex]a_0=3[/tex]. However,
[tex]a_1=2\neq-\dfrac3{1!}=-3[/tex]
but we can adjust for this easily:
[tex]y(x)=-3+2x-\dfrac3{2!}x^2-\dfrac3{3!}x^3-\dfrac3{4!}x^4+\cdots[/tex]
[tex]y(x)=5x-3-3x-\dfrac3{2!}x^2-\dfrac3{3!}x^3-\dfrac3{4!}x^4+\cdots[/tex]
Now all the terms following [tex]5x[/tex] resemble an exponential series:
[tex]y(x)=5x-3\displaystyle\sum_{n\ge0}\frac{x^n}{n!}[/tex]
[tex]\implies\boxed{y(x)=5x-3e^x}[/tex]
