$\frac{d^2 y}{d x^2}-(x-1) \frac{d y}{d x}=0$

$(x-1) \frac{d^2 y}{d x^2}-\frac{d y}{d x}=0$

$\frac{d^2 y}{d x^2}+(x-1) \frac{d y}{d x}-y=0$

$(x-1) \frac{d^2 y}{d x^2}+\frac{d y}{d x}=0$

$y=x e^x+c$

$y=x e^x+c e^{-x}$

$y=\frac{e^x+c}{x}$

$y=\frac{e^{-x}+c x}{x}$

$\log x+\tan \frac{y}{x}=C$

$\log x+\cos \frac{y}{x}=C$

$\log x-\sin \frac{y}{x}=C$

$\log x-\cos \frac{y}{x}=C$