Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 81 / 4 / ii / a / Solution

For positive $\lambda ,\alpha$ and $0<s<1$, the constant term of the quadratic equation is negative. Its roots are therefore real with opposite signs, so one has positive growth. At $s>1$, both the coefficient of $\beta$ and the constant term are positive. If the roots are real they are both negative; if complex they have negative real parts. Thus the growing band remains $\boxed{0<k^2{a}^2<1}$. At $s=0$ or $1$ there is a neutral root. If $\alpha =0$, the result reduces continuously to the clean-film capillary branch with a passive zero root.