= Solution
The <direction field> has slope $1-y^2$, independent of $x$. Its line segments slope upward for $-1<y<1$, are horizontal at $y=\pm1$, and slope downward for $|y|>1$. Thus $y=1$ is an attracting <equilibrium> in forward $x$, while $y=-1$ is repelling. The two constant solutions $y\equiv\pm1$ must be included separately when separating variables.
The following original sketch shows the <slope field> and one solution in each requested region. The lower exterior solution ends at a finite pole; it cannot be continued across that pole as a real finite-valued solution.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/ia/paper-2-slope-field.png]
{title=Direction field, equilibrium lines and representative solutions in the three regions of the autonomous scalar equation}
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