91Ó°ÊÓ

First-order differential equations are equations that relate a function, its derivatives, and possibly an independent variable. In these equations, only the first derivative of the function appears. The general form is \( \frac{dy}{dt} = f(t, y) \), where \( f(t, y) \) is some function of \( t \) and \( y \). These equations can describe a wide range of real-world phenomena, from population growth to mixing problems.

They can be solved using various methods like:

• Separation of variables: This involves rearranging the equation to allow the integration of one side with respect to \( y \) and the other with respect to \( t \).
• Integrating factors: A clever multiplication of the entire equation, that simplifies solving it.

For instance, in the exercise, our differential equation is \( \frac{dV}{dt} = 1.2 - 0.008V \), a first-order linear equation that models the change in volume of nitric acid over time. Solving it involves finding the function \( V(t) \), which represents the volume of nitric acid at any given time \( t \). This is crucial as it allows us to trace the behavior of a system's dynamics.

Mixing problems are a fascinating application of first-order differential equations. These problems involve a solution that is continuously being mixed, with specific rates of inflow and outflow. The aim is typically to find the concentration of a particular substance in the solution over time. These are classic problems in chemical and environmental engineering, where understanding concentrations is critical.

In these problems:

• The rate at which substances enter the tank is calculated as the product of the concentration and the flow rate.
• The rate at which substances leave the tank is determined by the outflow rate times the concentration of the substance in the tank.

In our scenario, nitric acid enters the tank at a rate of \( 1.2 \text{ L/min} \) and leaves proportional to the amount present, modeled as \( 0.008V(t) \). These conditions allow us to set up the differential equation \( \frac{dV}{dt} = 1.2 - 0.008V \). Solving this communicates how the concentration changes over time. The provided solution applies this model effectively, determining that the volume of nitric acid changes according to \( V(t) = 150 + 50e^{-0.008t} \).

Exponential decay is a concept where a quantity decreases at a rate proportional to its current value. This phenomenon appears in various natural and applied fields, from radioactive decay to depreciation of assets. In the context of differential equations, an exponential decay model can be neatly expressed in the form: \( N(t) = N_0 e^{-kt} \), where \( N_0 \) is the initial quantity, \( k \) is the decay constant, and \( t \) is time.

In our exercise, the term \( 50e^{-0.008t} \) represents an exponential decay model. It is part of the solution to the differential equation describing the volume of nitric acid. Decay models help us understand how the concentration decreases due to outflow and mixing.

Overall, using exponential decay alongside first-order differential equations and mixing problem frameworks allows us to predict when certain concentration thresholds will be reached, such as the 10% concentration in our exercise. Understanding these models is vital to predict and control behaviors in dynamic systems accurately.