I have this IVP: \begin{align*} \dot{x}_1(t)&=\frac{f_3}{V_3}\,x_3(t)-\frac{f_1}{V_1}\,x_1(t)+f(t)\\ \dot{x}_2(t)&=\frac{f_1}{V_1}\,x_1(t)-\frac{f_2}{V_2}\,x_2(t)\\ \dot{x}_3(t)&=\frac{f_2}{V_2}\,x_2(t)-\frac{f_3}{V_3}\,x_3(t). \end{align*} In particular, take $f_i/V_i=0.001,\;i=1,2,3$ and let $f(t)=0.125\;\text{lb/min}$ for the first $48$ hours, thereafter, $f(t)=0.$ Use the Laplace transform to solve for $x_1(t).$
I don't know how to proceed with this problem. The examples shown in our class always give the initial value of the variable to look for and its derivative. How do I solve this problem if the initial value given is not the variable I'm looking for?
Íf not stated explicitly differently, in this type of tasks all functions are zero at negative times, all inputs and reactions to them start only at time $t=0$. Usually this is expressed as $$x_k(0^-)=\lim_{t<0,\,t\to 0}x_k(t)=0.$$
The units given for the constants and function values do not match, if the $x_k$ are in [lb] and $t$ in [min], then the derivative matches with $f$, but $f_i/V_i$ should have unit [min$^{-1}$].