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Equilibrium solutions are a fundamental concept in differential equations. They occur when the system is in a state where no changes happen over time. In mathematical terms, for a given differential equation \( \frac{dy}{dt} = f(y) \), the equilibrium solution is found by setting \( f(y) = 0 \). This translates to finding the values of \( y \) where the system is "at rest".
For example, consider the equation \( \frac{dy}{dt} = y - 1 \). The equilibrium solution is determined by solving \( y - 1 = 0 \), leading to \( y = 1 \). This means that when \( y \) is 1, there is no change in the system, marking it as an equilibrium point. Similarly, for the equation \( \frac{dC}{dt} = \frac{F}{V} c_{i} - \frac{F}{V} C \), setting it to zero gives \( C = c_{i} \), indicating that at this point, the concentration remains constant over time.
Equilibrium solutions provide valuable insights into the behavior of systems modeled by differential equations, revealing conditions under which the system remains constant.

Stability analysis is a crucial step to understand how a system behaves when disturbed from its equilibrium state. Once we determine the equilibrium solution, the next step is to check this solution's stability by analyzing the derivative of the function.
To do this, we find the derivative \( f'(y) \) of the original function \( f(y) \). The nature of this derivative at the equilibrium point can reveal whether the solution is stable or unstable:

• If \( f'(y) < 0 \), the equilibrium is stable, meaning the system returns to equilibrium after a small disturbance.
• If \( f'(y) > 0 \), the equilibrium is unstable, meaning the system moves away from the equilibrium after a disturbance.

In Part (a) of our example, the function is \( f(y) = y - 1 \), thus \( f'(y) = 1 \). Since \( f'(1) > 0 \), the equilibrium \( y = 1 \) is unstable, indicating that any small deviation will cause the system to diverge from this point.
In Part (b), the rewritten equation shows \( f'(C) = -\frac{F}{V} \). Since this is negative, the equilibrium solution \( C = c_{i} \) is stable, meaning if the system is disturbed, it will naturally return to its equilibrium state.

Mathematical modelling is the process of using mathematical structures and concepts to represent real-world systems. Differential equations often play a key role in these models due to their ability to describe how systems change over time.
For example, the differential equation \( \frac{dC}{dt} = \frac{F}{V} c_{i} - \frac{F}{V} C \) models how the concentration \( C \) changes with time based on the inflow and outflow rates in a vessel. Here, \( F \), \( V \), and \( c_{i} \) are constants reflecting the system's properties. This model helps predict how the concentration will adjust based on different parameters and conditions.
Similarly, the equation \( \frac{dy}{dt} = y - 1 \) can model simple biological processes or physical systems where growth or decay moves towards a particular value (in this case, 1).

• Through mathematical modelling, complex processes become analyzable and predictable.
• It aids in decision-making and system optimization by providing insights into behavior under various scenarios.

These models are crucial for scientists and engineers as they simplify understanding complex systems and allow for effective predictions and solutions.