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Separation of variables is a powerful technique used to solve differential equations. It is applied when you can rearrange a differential equation so that each variable, along with its differential, appears on opposite sides of the equation. This method is particularly useful for first-order differential equations, such as the one in this example.

Let's break this down: we start with \( \frac{dy}{dx} = \left(\frac{2y+3}{4x+5}\right)^2 \). We aim to set all terms involving 'y' on one side and all terms involving 'x' on the other. By manipulating the equation, we achieve \( \frac{1}{(2y+3)^2} \, dy = \frac{1}{(4x+5)^2} \, dx \).

This arrangement lets us handle the 'y' terms separately from the 'x' terms. The method has mobilized complex differential equations by reducing them to a form where both sides can be separately integrated, as seen here.

A first-order differential equation is one that involves the first derivative of a function but no higher derivatives. The problem we are examining is a textbook example of a first-order differential equation: the equation \( \frac{dy}{dx} = \left(\frac{2y+3}{4x+5}\right)^2 \) contains only the first derivative \( \frac{dy}{dx} \).

First-order differential equations describe rates of change and appear frequently in fields like physics, biology, and economics. They can often be solved using various methods, including the separation of variables. Solutions to these equations provide a function 'y' that describes how 'y' changes relative to 'x'. This characteristic makes them incredibly useful for modeling natural phenomena.

Integration is a fundamental concept in calculus used to solve differential equations. Once we have separated variables, as in \( \frac{1}{(2y+3)^2} \, dy = \frac{1}{(4x+5)^2} \, dx \), each side of the equation is integrated separately. Integration reverses differentiation, allowing us to find a function from its derivative.

For the left side, integrate with respect to 'y':
- \( \int \frac{1}{(2y+3)^2} \, dy \) uses substitution or recognition as a standard form, resulting in \( -\frac{1}{2(2y+3)} + C_1 \).

For the right side, integrate with respect to 'x':
- \( \int \frac{1}{(4x+5)^2} \, dx \), which by substitute or recognizing yields \( -\frac{1}{4(4x+5)} + C_2 \).

Integrating both sides transforms a complicated rate of change to simpler expressions, leading us to solve the initial problem.

Solving differential equations involves finding a function or set of functions that satisfy the differential equation. Once we integrate both sides of a separated differential equation, it's essential to combine the constants obtained from integration into a single constant:\( C \). This process gives us the general solution.

In our exercise, combining constants leads us to \( \frac{1}{2(2y+3)} = \frac{1}{4(4x+5)} + C \). Solving for 'y' in terms of 'x' provides the particular solution, which is often the ultimate goal.

The beauty of solving differential equations is that it enables us to predict and understand dynamic systems. Whether it's predicting population growth or calculating electrical currents, this solution equips us with a mathematical model describing the system's behavior.