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A differential equation is a mathematical equation that involves functions and their derivatives. It tells us how one variable changes with respect to another. In simpler terms, it's an equation that involves rates of change. Think about how the speed of your car changes over time; that's a real-world example of a differential equation at work. In our exercise, the equation \(\frac{dQ}{dt} = k(Q-70)\) represents how the quantity \(Q\) changes over time \(t\). Since it involves both the function \(Q\) and its derivative with respect to time, it is classified as a differential equation.Differential equations are extremely powerful as they can describe various physical phenomena, such as motion, growth, decay, and more. By solving these equations, we can predict future behavior or understand underlying processes.

Separation of variables is a technique used to simplify and solve differential equations. It involves rearranging an equation so that each variable appears on its own side of the equation. The goal is to isolate the dependent variable (for example, \(Q\) in our exercise) and its derivatives on one side of the equation. Meanwhile, move the independent variable (like \(t\), in this case) to the other side. Once the variables are separated, you can integrate each side independently. In our example, we separate the equation \(\frac{dQ}{dt} = k(Q-70)\) into \(\frac{dQ}{Q-70} = k \, dt\). This makes it easier to solve because we can now focus on integrating each part separately.

Integration is a fundamental concept in calculus. It is the process of finding the integral of a function, which can be understood as the reverse of differentiation. In the context of differential equations, integration helps us find a function from its derivative. After separating the variables in a differential equation, the next step is to integrate both sides. In our example, we integrate \(\int \frac{1}{Q-70} \, dQ\) and \(\int k \, dt\). The integration involves finding antiderivatives, resulting in expressions like \[\ln |Q-70| = kt + C\]. Here, \(C\) is called the constant of integration. Integration can solve problems related to area under curves, total change, and more, making it versatile and essential in mathematical analysis.

Exponential growth and decay are processes described by specific types of differential equations, usually involving continuous change. In exponential growth, the rate of change of a quantity is directly proportional to the quantity itself. This leads to rapid increase over time. Conversely, exponential decay refers to processes where the quantity decreases over time, such as radioactive decay.In our exercise solution, we reached \(Q = Ce^{kt} + 70\), an equation that describes how \(Q\) behaves over time. If \(k > 0\), the function exhibits exponential growth. If \(k < 0\), it demonstrates exponential decay. This concept is widely applied in science and finance, such as predicting population growth or calculating compound interest. Understanding exponential behavior helps in predicting how systems evolve over time.