Yes, the provided solution steps seem to be correct for solving the given differential equation. Here's an explanation of what is happening step-by-step:

1. Original Differential Equation: $\frac{di}{ds} = \frac{\frac{di}{dt}}{\frac{ds}{dt}} = \frac{gsi - fi}{-gsi}$ This establishes the relationship between $\frac{di}{ds}$ and $\frac{di}{dt}, \frac{ds}{dt}$, using the chain rule. The equation is simplified to: $\frac{di}{ds} = -1 + \frac{f}{gsi}.$

2. Separate Variables: Rearrange terms: $di = -1 \, ds + \frac{f}{gs} \, ds.$

3. Integration: Integrate both sides: $i = \int -1 \, ds + \int \frac{f}{g} \frac{1}{s} \, ds.$ The first integral evaluates to $-s$, and the second evaluates to $\frac{f}{g} \ln s$, resulting in: $i = -s + \frac{f}{g} \ln s + C,$ where $C$ is the integration constant.

This approach aligns with standard techniques for solving first-order separable differential equations. The result is correct.

Would you like further clarification, or is there a specific step you'd like me to expand on?

Follow-up Questions:

1. Why does the chain rule apply to connect $\frac{di}{dt}$ and $\frac{ds}{dt}$?
2. Can you explain why $-\int ds = -s$ holds for the first integral?
3. What are the assumptions about $g$ and $s$ in the logarithmic term $\ln s$?
4. How can the solution be checked for correctness after integrating?
5. What is the role of the constant $C$, and how can it be determined in a specific context?

Tip:

When solving differential equations, always check the domain of the solution (e.g., ensuring $s > 0$ for $\ln s$ to be defined).