Chapter 10.3, Problem 37E

When a nonlinear capacitor is present in an LRC-circuit, the voltage drop is no longer given by q/C but is more accurately described by αq + βq³, where α and β are constants and α > 0. Differential equation (34) of Section 5.1 for the free circuit is then replaced by

$L\frac{d^{2}q}{d{t}^2}+R\frac{dq}{dt}+\alpha q+\beta q^3=0.$

Find and classify all critical points of this nonlinear differential equation. [Hint: Divide into the two cases β > 0 and β < 0.]