Let’s consider the second order linear nonhomogeneous differential equation \((x-1)y\,^{\prime \prime }-x y\,^{\prime }+y=(x-1)^2e^{x}\).

(a)
Given that \(y_1=e^x\) is a solution to \((x-1)y\,^{\prime \prime }-x y\,^{\prime }+y=0\) let’s find the general solution to \((x-1)y\,^{\prime \prime }-x y\,^{\prime }+y=(x-1)^2e^{x}\) by Reduction of Order.
(b)
Let’s solve the IVP \((x-1)y\,^{\prime \prime }-x y\,^{\prime }+y=(x-1)^2e^{x}\), \(y(2)=0\), \(y^{\,\prime }(2)=0\).