Chapter 1 - First-Order Differential Equations - 1.11 Some Higher-Order Differential Equations - Problems - Page 105: 3

$y(x)=c \ln |x-1|+C_1x+C_2$

Suppose that $a= \dfrac{dy}{dx}$ and $ \dfrac{da}{dx}= \dfrac{d^2y}{dx^2}$ Integrating factor; $I.P.=e^{\int -\frac{1}{(x-1)(x-2)} dx}=\dfrac{x-1}{x-2}$ Now, $\dfrac{d}{dx}(a\dfrac{x-1}{x-2})=\dfrac{1}{(x-2)^2}$ Integrate to obtain: $a\dfrac{x-1}{x-2}=-\dfrac{1}{x-2}+C \implies \dfrac{dy}{dx}=-\dfrac{1}{x-2}+C-\dfrac{x-2}{x-1}$ Therefore, the general solution is: $y(x)=c \ln |x-1|+C_1x+C_2$