I am stuck with a homework problem which is about the modelling of tumor growth by ODEs.
The function $A(t)$ is the amount of drugs in the patients blood. During some intervals (namely, $(n,n+\tau)$, the drug is added with an infusion, and in $(n+\tau,n+1)$ it is only used up by the body (\tau<1).
The function that then describes this is given by:
$A(t)=\begin{cases} \frac{a_*}{\lambda}+\left( A_n-\frac{a_*}{\lambda} \right) e^{-\lambda(t-n)}, & \text{for }n\leq t \leq n+\tau \\ \left[ \frac{a_*}{\lambda}+\left( A_n-\frac{a_*}{\lambda}\right)e^{-\lambda \tau}\right] e^{-\lambda(t-n-\tau)} & \text{else} \end{cases}$
And the task is to check that this is true. This includes checking for continuity at $t=n, t=n+\tau$. So when I plug in $n+\tau$, I get $\frac{a_*}{\lambda}+ \left( A_n-\frac{a_*}{\lambda}\right)e^{-\lambda \tau}$ on both sides, but at $t=n$ I get only: $\frac{a_*}{\lambda}+A_n-\frac{a_*}{\lambda}=e^{\lambda \tau}\frac{a_*}{\lambda}+A_n-\frac{a_*}{\lambda}$, which is not equal obviously. The things is, the function $A(t)$ is given (I did not have to find it, just check it), so there should not be a mistake. But where else is the problem?
Also, on a sidenote, the problem asks me to draw a circuit diagram of the ODE model $N(t),A(t)$, where $N$ is the # of cells in the tumor. What exactly is a circuit diagram? Would that be the same as a network diagram?