Answer: Suppose an RC-series circuit has a variable resistor. If the resistance at time

Chapter 2, Problem 37

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Suppose an RC-series circuit has a variable resistor. If the resistance at time t is defined by \(R(t)=k_{1}+k_{2} t\), where \(k_{1}\) and \(k_{2}\) are known positive constants, then the differential equation in (10) of Section 2.7 becomes

\(\left(k_{1}+k_{2} t\right) \frac{d q}{d t}+\frac{1}{C} q=E(t)\),

where C is a constant. If \(E(t)=E_{0}\) and \(q(0)=q_{0}\), where \(E_{0}\) and \(q_{0}\)  are constants, then show that

\(q(t)=E_{0} C+\left(q_{0}-E_{0} C\right)\left(\frac{k_{1}}{k_{1}+k_{2} t}\right)^{1 / C k_{2}}\).