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The equation provided is a first-order nonlinear differential equation of the form:

\[
\frac{dy}{dt} = sy(a - y)(y - b),
\]

where \( s \), \( a \), and \( b \) are constants.

### Steps to Solve:

This is a separable differential equation, which means we can separate the variables \( y \) and \( t \) on opposite sides of the equation.

1. **Separate the variables**: Move all terms involving \( y \) to one side and the differential term involving \( t \) to the other.
   \[
   \frac{1}{y(a - y)(y - b)} dy = s \, dt
   \]
   
2. **Integrate both sides**: To solve, integrate both sides with respect to their respective variables:
   - The left side involves a complex rational function, so partial fraction decomposition may be needed to simplify the expression for integration.
   - The right side is straightforward and will integrate to \( st + C \), where \( C \) is the constant of integration.

Do you want me to proceed with the detailed integration and solution of this equation? Let me know if you have any specific questions! 

### Related Questions:
1. How do you apply partial fraction decomposition to the left-hand side of this equation?
2. Can this type of differential equation model real-world systems?
3. What happens if \( a = b \)? How does the solution change?
4. How can you solve the equation numerically if exact integration is difficult?
5. What is the behavior of the solution as \( t \to \infty \)?

**Tip**: Always check for equilibrium points in a differential equation, which occur when \( \frac{dy}{dt} = 0 \). This can give insight into the system's long-term behavior.

The query submitted by the user: dy/dt = sy(a - y)(y - b), where s, a, b are constants.

Learn how to solve the nonlinear separable differential equation dy/dt = sy(a - y)(y - b). This problem explores key mathematical concepts like separation of variables, partial fraction decomposition, and equilibrium points. Suitable for college-level students.

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