Answer
$Q(t)=-0.02e^{-10t}[\cos 10t + \sin 10t]+0.03$
Work Step by Step
After plugging the data in, we get $\dfrac{d^2 Q}{dt^2}+20\dfrac{dQ}{dt}+200Q=6$
The auxiliary equation is: $r^2+20r+200=0$
$\implies r= -10 \pm 10i$
Here,we have $Q_c= e^{-10t}(\sin 10t +\cos 10t)$
The particular solution is: $Q_p=A$ and $ Q'_p=0; Q''_p=0$
$A=\dfrac{3}{100}$ (Simplify)
The particular solution is: $Q_p=\dfrac{3}{100}$
Now, $Q(t)=e^{-10t}(c_1\sin 10t +c_2\cos 10t)+\dfrac{3}{100}$
Consider at $t=0$
$Q(0)=e^{-10t}(c_1\sin 10t +c_2\cos 10t)+\dfrac{3}{100} \implies \dfrac{1}{100}=e^{0}(c_1\sin 0 +c_2\cos 0)+\dfrac{1}{100}$
This yields: $c_1=-\dfrac{1}{50}=-0.02 ; c_2=-\dfrac{1}{50}=-0.02$
Hence, $Q(t)=-0.02e^{-10t}[\cos 10t + \sin 10t]+0.03$
