Variable Resistor: In Section $5\mathrm{.}1($page $208$ Formula $(34)),$ described by a linear second$-$order differential equation wiht charge $q$ on the capacitor in a simple $RLC$ circuit is governed by the equation.

$L{q}^{\text{'}\text{'}}(t)+R{q}^{\text{'}}(t)+\frac{1}{C}q(t)=E(t)$

where $L$ is the inductance, $R$ the resistance, $C$ the capacitance, and $E$ the voltage source. Since the resistance of a resistor increases with temperature, let's assume that the resistor is heated so that the resistance at time $t$ is $R(t)=1+\frac{1}{10}t.$ If $L=\frac{1}{10}H,C=2F,E(t)=0V,q(0)=10C$ and $q^{\text{'}}(0)=0A,$ find at least the first four nonzero terms in a power series expansion about $t=0$ for the charge on the capacitor.