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A **First-order linear differential equation** is a type of differential equation that involves the first derivative of the unknown function, usually denoted as \( y' \), and the function itself, \( y \). The general form is

• \( y' + P(x)y = Q(x) \)

Among the important characteristics of such equations are:

• They only involve the first derivative \( y' \) of the function \( y(x) \).
• Both \( P(x) \) and \( Q(x) \) are functions solely of \( x \). This linearity with respect to \( y \) and its derivatives is what qualifies them as "linear."

These equations are quite common due to their simplicity and the intuitive physical phenomena they can model. To solve them practically, you often need to convert it into a form where the solution can be readily derived. This preparatory step is crucial to utilize techniques like finding an integrating factor.
In our problem, dividing the original equation by \( x^3 \) puts it into this standard recognizable form.

Solving **first-order linear differential equations** often requires an integrating factor, which is a function that simplifies the solution process. The integrating factor \( \mu(x) \) is calculated from the function \( P(x) \) found in the differential equation:

• For example, if \( y' + \frac{3}{x}y = \frac{5}{x^3} \sinh(10x) \), then \( P(x) = \frac{3}{x} \).
• The integrating factor is then \( \mu(x) = e^{\int P(x) \, dx} = e^{3\ln |x|} = x^3 \).

The purpose of multiplying the entire differential equation by this integrating factor is to convert the equation into an "exact" form, which allows for straightforward integration. After this transformation, the equation can be rewritten so that the left-hand side is the derivative of \( x^3 y \). This simplification makes it much easier to integrate both sides with respect to \( x \).
Once we perform this operation, solving the equation becomes a matter of simple integration over both sides, leading directly to the solution in terms of \( y \).

You might have noticed the term involving **hyperbolic functions** in the equation: \( 5 \sinh(10x) \). Hyperbolic functions, such as \( \sinh \) and \( \cosh \), relate closely to exponential functions:

• The hyperbolic sine function is defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \).
• The hyperbolic cosine function is similarly defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).

These functions often appear in solutions to differential equations, especially when modeling phenomena involving hyperbolic geometry or certain types of non-linear systems. In this exercise, integration involving the hyperbolic sine function was necessary to solve the differential equation fully.
When integrating hyperbolic functions, they often translate smoothly into their exponential form, making them relatively straightforward to handle in calculus. For instance, integrating \( 5 \sinh(10x) \), we find the result \(-\frac{1}{2}\cosh(10x) + C\), which is easier to manage when expressed using \( \cosh \).
Understanding these functions' relationships and properties can greatly simplify working with complex differential equations like this one.