91Ó°ÊÓ

Differentiation is a process in mathematics used to find the rate at which a quantity changes. This process, known as taking the derivative, plays a key role in understanding how functions behave. When we differentiate a function, we calculate the slope of its graph at any given point.

In our specific exercise, the function provided is: \[x = A \mathrm{e}^{-2t}\]. Differentiating this with respect to the variable \(t\), we can find how \(x\) changes as \(t\) changes.

- The derivative of \(\mathrm{e}^{-2t}\) with respect to \(t\) is \(-2 \mathrm{e}^{-2t}\) because of the chain rule.- Multiplying by the constant \(A\), the derivative becomes \(-2A \mathrm{e}^{-2t}\).Thus, the rate of change of \(x\) with respect to time \(t\) is given by \( \frac{dx}{dt} = -2A \mathrm{e}^{-2t} \). Differentiation helps us find this relationship, which is crucial for developing our differential equation.

An exponential function is a mathematical function of the form \(f(t) = a \cdot \mathrm{e}^{kt}\), where \(a\) and \(k\) are constants, and \(e\) is the base of natural logarithms, approximately equal to 2.71828.

Exponential functions are common in modeling real-world phenomena involving growth or decay, such as population growth, radioactive decay, and in this case, a decaying function of time. The hallmark of exponential functions is that they change proportionally to their current value.

In our exercise, we have \(x = A \mathrm{e}^{-2t}\), meaning as time \(t\) increases:

• The value of \(x\) decreases.
• The negative exponent \(-2t\) indicates exponential decay.

Exponential decay means the function value gets smaller at a rate relative to its size—a fundamental concept in differential equations where natural dynamics involve rates of change.

Separation of variables is a technique used to solve first-order differential equations, where the variables can be isolated on opposite sides of the equation.

For the equation \(\frac{dx}{dt} + 2x = 0\), we can rearrange terms to separate the function of \(x\) from the function of \(t\).

- First, by moving \(2x\) to the other side, we get \(\frac{dx}{dt} = -2x\).- This allows us to express the relationship with differentials: \(\frac{1}{x}dx = -2 dt\).The principle of separation of variables means each side of this equation can be integrated independently:

• Integrate \(\frac{1}{x} dx\) to get \(\ln|x| + C_1\).
• Integrate \(-2 dt\) to get \(-2t + C_2\).

By isolating and integrating the variables, we can then continue to solve for \(x(t)\), revealing how the process of separation helps in crafting solutions for differential equations.