This is all fine.
Course: Self- Study
Textbook: A Course in Mathematical Biology
I'm working with the basic SIR model, and I have this system:
$$\begin{align}
S'(t) &= -\beta S I \\
I'(t) &= \beta S I - \alpha I
\end{align}$$
We look for solutions in the form $I(S)$ so we divide $I'$ by $S'$ making $$\frac{dI}{dS} = \frac{\alpha}{\beta S} - 1$$
I have a question that says to determine the constant $C$ such that $I(S_0) = I_0$.
By solving by separation of variables, we see that $I(S) = \frac{\alpha}{\beta} \ln(S) - S + C$. So
$$\begin{align}
I(S_0) &= \frac{\alpha}{\beta} \ln(S_0) - S_0 + C = I_0 \\
C &= S_0 + I_0 - \frac{\alpha}{\beta} \ln(S_0)
\end{align}$$
In my chapter there have not been any initial conditions, so I'm wondering if leaving C in terms of $S_0$ and $I_0$ would be fine, because I'm supposed to create a Matlab function for $I(S)$ with the appropriate constant of integration.