When a nonlinear capacitor is present in an LRC circuit the voltage drop is no...

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When a nonlinear capacitor is present in an LRC circuit the voltage drop is no longer given by g C but is more accurately described by aq q where a and are constants and a 0 Differential equation 34 of Section 5 1 for the free circuit is then replaced by Lda Rdg ag Bq 0 d q dt dt Find and classify all critical points of this nonlinear differential equation Hint Divide into the two cases 0 and 0 Assume x q y q and R 4a Order your answers from smallest to largest x then from smallest to largest y Assuming 0 one obtains the following real critical point X Conclusion x y 0 0 Assuming 0 one obtains the following real critical points X Conclusion x y x y x y a R 0 0 0 0 0 X stable node or spiral point saddle point stable node or spiral point saddle point

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When a nonlinear capacitor is present in an LRC circuit the voltage drop is no longer given by g C but is more accurately described by aq q where a and are constants and a 0 Differential equation 34 of Section 5 1 for the free circuit is then replaced by Lda Rdg ag Bq 0 d q dt dt Find and classify all critical points of this nonlinear differential equation Hint Divide into the two cases 0 and 0 Assume x q y q and R 4a Order your answers from smallest to largest x then from smallest to largest y Assuming 0 one obtains the following real critical point X Conclusion x y 0 0 Assuming 0 one obtains the following real critical points X Conclusion x y x y x y a R 0 0 0 0 0 X stable node or spiral point saddle point stable node or spiral point saddle point