y′=y2−1y' = y^2 - 1y′=y2−1 dydx=y2−1\frac{dy}{dx} = y^2 - 1dxdy=y2−1 ∫1y2−1dy=∫dx\int \frac{1}{y^2-1} dy = \int dx∫y2−11dy=∫dx 12log∣y−1y+1∣=x+C1\frac{1}{2}\log\left|\frac{y-1}{y+1}\right| = x + C_121logy+1y−1=x+C1 e2C1=Ce^{2C_1} = Ce2C1=C とおくと y−1y+1=Ce2x\frac{y-1}{y+1} = Ce^{2x}y+1y−1=Ce2x 1−2y+1=Ce2x1 - \frac{2}{y+1} = Ce^{2x}1−y+12=Ce2x 2y+1=1−Ce2x\frac{2}{y+1} = 1 - Ce^{2x}y+12=1−Ce2x y=21−Ce2x−1y = \frac{2}{1-Ce^{2x}} - 1y=1−Ce2x2−1 y(0)=−1y(0) = -1y(0)=−1 のとき C=−∞C = -\inftyC=−∞ よって y=−1y = -1y=−1 y(0)=0y(0) = 0y(0)=0 のとき C=−1C = -1C=−1 よって y=1−e2x1+e2xy = \frac{1-e^{2x}}{1+e^{2x}}y=1+e2x1−e2x y(0)=1y(0) = 1y(0)=1 のとき C=0C = 0C=0 よって y=1y = 1y=1 y(0)=2y(0) = 2y(0)=2 のとき C=13C = \frac{1}{3}C=31 よって y=3+e2x3−e2xy = \frac{3+e^{2x}}{3-e^{2x}}y=3−e2x3+e2x