91Ó°ÊÓ

A first-order system in differential equations involves relations where the derivatives are first order, meaning the highest derivative is to the first degree. Such systems are commonly used to describe real-world phenomena, like motion or population growth. For a system like the one in this exercise, there are two equations:

• \( \frac{dx}{dt} = y^2 \)
• \( \frac{dy}{dt} = xy \)

In these equations, \(x\) and \(y\) change over time, and the equations describe how their rates of change are interconnected. Solving these systems often involves techniques from calculus, such as differentiation. In this particular problem, the objective is to check if a given pair of functions can satisfy these change relationships.

The hyperbolic tangent function, denoted as \( \tanh \), is similar to the tangent function in trigonometry but applies to hyperbolic functions. It's defined as:\[ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \]This function is notable for its S-shaped curve, which approaches -1 as \(x\) goes to negative infinity, and 1 as \(x\) goes to positive infinity. In practice, \( \tanh \) is often used in areas like neural networks as an activation function due to its ability to bring inputs into a normalized range.
In our differential equation context here, it's part of the function defining \( x = b \tanh(bt) \). Differentiating this function is crucial in checking if it satisfies the first-order system.

The hyperbolic secant function, \( \text{sech} \), is defined as the reciprocal of the hyperbolic cosine function, \( \cosh \). Its definition is:\[ \text{sech}(x) = \frac{1}{\cosh(x)} \]This function is limited between 0 and 1, reaching its maximum at \(x = 0\). It's a key player in defining function \(y\) in this exercise:
\( y = b \text{sech}(bt) \).
When we square \(y\), the function \(\text{sech}^2(bt)\) comes into play, which is integral in meeting the first differential equation requirement, \( \frac{dx}{dt} = y^2 \). This validation is part of showing that the function pair is a valid solution to the system.

The chain rule is a fundamental differentiation rule used in calculus for finding the derivative of a composite function. In simple terms, if you have a function \(f(g(x))\), the derivative is:\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]This technique allows us to break down complex functions into simpler parts for differentiation.

• For \( x = b \tanh(bt) \), the derivative involves the derivative of \( \tanh(u) \) where \( u = bt \).
• For \( y = b \text{sech}(bt) \), the differentiation similarly involves using the chain rule, resulting in multiplying by the derivative of the inside function \( bt \).

Understanding and applying the chain rule accurately is crucial in solving the given differential equations, ensuring that the obtained functions correctly describe the system's behavior over time.