## anonymous 4 years ago Solve separable differential equation: <see attached>

1. anonymous
2. anonymous

You have that $\frac{dy}{dx}=xy^2-x-y^2+1=x(y^2-1)-(y^2-1)=(x-1)(y^2-1)$ $\frac{dy}{y^2-1}=(x-1)dx$ Using partial fractions you can rewrite the left hand side as $\frac12\left(\frac{1}{y-1}-\frac{1}{y+1}\right)dy=(x-1)dx$ Integrating both sides then gives $\frac12(\ln|y-1|-\ln|y+1|)=\frac12x^2-x+C$ Multiplying both sides by 2 and combining the logs gives $\ln\left|\frac{y-1}{y+1}\right|=\ln\left|1-\frac{2}{y+1}\right|=x^2-2x+C$ Exponentiating each side gives $1-\frac{2}{y+1}=Ae^{x^2-2x}$ And then simplifying gives yours final answer of $y=\frac{2}{1-Ae^{x^2-2x}}-1$

3. anonymous

okay, I got stuck at the very first line with the factoring. Thank you so much.