Answer to Question #167515 in Differential Equations for Cecille

Question

dy/dx-y=xy²

Answer 1

$\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x} - y = xy^2 \\ y^{-2}\frac{\mathrm{d}y}{\mathrm{d}x} - y^{-1} = x \\ \textsf{Let}\,\,\, v = y^{-1}\\ \frac{\mathrm{d}v}{\mathrm{d}x} = -y^{-2}\frac{\mathrm{d}y}{\mathrm{d}x} \\ -\frac{\mathrm{d}v}{\mathrm{d}x} - v = x \\ \frac{\mathrm{d}v}{\mathrm{d}x} + v = -x \\ \textsf{Integrating factor (IF)} = e^{\int 1 \mathrm{d} x} = e^x\\ v e^{x} = -\int x e^{x} \,\, \mathrm{d}x \\ v e^{x} = -xe^{x} + \int e^{x}\,\, \mathrm{d}x \\ ve^{x} = -xe^{x} + e^{x} + C \\ v = 1 - x + Ce^{-x} \\ y^{-1} = 1 - x + Ce^{-x} \\ \therefore y = \frac{1}{1 - x + Ce^{-x}}$