Chapter 3, Problem 11RE

Suppose an RC-series circuit has a variable resistor. If the resistance at time t is defined by R(t) = k₁ + k₂t, where k₁ and k₂ are known positive constants, then the differential equation in (9) of Section 3.1 becomes

$(k_1+k_{2}t)\frac{dq}{dt}+\frac{1}{C}q=E(t)$,

where C is a constant. If E(t) = E₀ and q(0) = q₀, where E₀ and q₀ are constants, then show that

$q(t)=E_{0}C+\left(q_0-E_{0}C\right){\left(\frac{k_1}{k_1+k_{2}t}\right)}^{1\text{/}C{k}_2}$.