(c) Determine the general solution for the differential equation, \[ \frac{d r}{d s}=\frac{s r+2 r-s-2}{s r-3 r+s-3} \]

To solve the differential equation \[ \frac{d r}{d s}=\frac{s r+2 r-s-2}{s r-3 r+s-3}, \] we can start by rewriting it in a more manageable form. Notice that both the numerator and denominator can be factored or rearranged. After some manipulation, we can express it as: \[ \frac{d r}{d s} = \frac{(s + 2)r - (s + 2)}{(s - 3)r + (s - 3)}. \] From here, it's helpful to separate the variables. If we can isolate \( r \) and \( s \), we can integrate both sides accordingly. The next step is to cross-multiply and rearrange: \[ (s r - 3 r + s - 3) \frac{d r}{d s} = (s + 2)r - (s + 2). \] Expanding and simplifying gives us a clearer equation, leading to the characteristic form of: \[ \frac{(s+2)r - (s+2)}{(s-3)r + (s-3)} = \frac{d r}{d s}. \] At this point, you can integrate both sides with respect to \( s \), which might lead you to a logarithmic solution or an implicit function relating \( r \) and \( s \). Ultimately, the solution will likely involve constants of integration, leading to the general solution for \( r \) in terms of \( s \). Remember to check for any specific values that could cause undefined conditions, such as division by zero. This differential equation certainly puts your algebraic skill to the test! Solving it can reveal rich connections between \( r \) and \( s \) that are vital in fields like physics or economics.