= Solution
Rotate coordinates to $s=(x+y)/\sqrt2$ and $t=(x-y)/\sqrt2$. The <stream function> becomes
$$
\psi=\frac12\left((\alpha-\beta)s^2-(\alpha+\beta)t^2\right).
$$
For $\alpha<\beta$, both coefficients are negative, so every nonzero contour is a bounded ellipse surrounding the stationary origin. For $\alpha>\beta$, the form is indefinite: nonzero contours are unbounded hyperbolas, and the zero contour consists of straight separatrices. Consequently \b[all nonstationary <streamlines> are bounded exactly when $\alpha<\beta$]. The origin is itself a stationary bounded trajectory in every case; the assertion about boundedness concerns the nonstationary <streamlines>. The equality case is handled separately below. These are the <linear planar strain and rotation streamlines>.
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