 Solution

Rotate coordinates to $s=(x+y)/\sqrt{2}$ and $t=(x-y)/\sqrt{2}$. The stream function becomesFor $\alpha <\beta$, 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 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.