Solve \(\displaystyle (x-1)y''-xy'+y=x^2-2x+1 \).

The solution is \(\displaystyle y=C_1 x +C_2 e^{x} -x^2 -x -1\).

any hint or shortcut ?

My thoughts are to consider the particular solution \(\displaystyle y_p \) as a polynomial, after that we find the other solution by \(\displaystyle y_2 / y_p = \int y_{p}^{-2}\cdot \mu(x) dx \;\) where \(\displaystyle \mu\) is the integrating factor of equation \(\displaystyle -xy' +y=0\). The general solution is \(\displaystyle y=y_p +y_h +y_2\).

Can we continue this way?

The solution is \(\displaystyle y=C_1 x +C_2 e^{x} -x^2 -x -1\).

any hint or shortcut ?

My thoughts are to consider the particular solution \(\displaystyle y_p \) as a polynomial, after that we find the other solution by \(\displaystyle y_2 / y_p = \int y_{p}^{-2}\cdot \mu(x) dx \;\) where \(\displaystyle \mu\) is the integrating factor of equation \(\displaystyle -xy' +y=0\). The general solution is \(\displaystyle y=y_p +y_h +y_2\).

Can we continue this way?

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