dy/dx-y=xy²
dydx−y=xy2y−2dydx−y−1=xLet v=y−1dvdx=−y−2dydx−dvdx−v=xdvdx+v=−xIntegrating factor (IF)=e∫1dx=exvex=−∫xex dxvex=−xex+∫ex dxvex=−xex+ex+Cv=1−x+Ce−xy−1=1−x+Ce−x∴y=11−x+Ce−x\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}}dxdy−y=xy2y−2dxdy−y−1=xLetv=y−1dxdv=−y−2dxdy−dxdv−v=xdxdv+v=−xIntegrating factor (IF)=e∫1dx=exvex=−∫xexdxvex=−xex+∫exdxvex=−xex+ex+Cv=1−x+Ce−xy−1=1−x+Ce−x∴y=1−x+Ce−x1
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