I'm trying to solve a system of equations: $$q_s[a+sK_o(\sqrt{bs})]+q_o[sK_o(\sqrt{cs})]=d$$ $$q_s[sK_o(\sqrt{cs})]+q_o[a+sK_o(\sqrt{es})]=f$$ I've used Cramer's rule to obtain a solution for $q_s$ and $q_o$ that works well in Laplace space. I'd like to have a time domain solution as well. I had thought that it might work if I did an inverse transform on each of the individual components of the equations but this doesn't seem to work. If I apply the inverse to $sK_o(\sqrt{bs})$ ( Inverse Laplace Transform for $ sK_0(\sqrt{as}) $?), then I also have to inverse the independent constants and they "drop out" (i.e $L^{-1}(c)=c\delta(t)=0,$ for $t>0$). I then thought maybe dividing through by $s$ would be a good approach: $$q_s[a/s+K_o(\sqrt{bs})]+q_o[K_o(\sqrt{cs})]=d/s$$ $$q_s[K_o(\sqrt{cs})]+q_o[a/s+K_o(\sqrt{es})]=f/s$$
The inverse transform of each term is readily obtained but I still don't get the correct $q_s$ or $q_o$. I suspect now that the issue must be that effectively $q_s$ and $q_o$ are functions rather than variables (i.e. $q_s(s)$ rather than $q_s$), so my inversions would need to take that into account.
Is someone able to help me solve for $q_s$ and $q_o$ in the time domain?