91Ó°ÊÓ

A first-order differential equation involves derivatives of the first degree, meaning it includes terms such as \( dx \) and \( dy \) but not higher-order derivatives like \( d^2x \) or \( d^2y \). It is pivotal because it simplifies our approach to solving problems involving rate and change.
First-order differential equations usually have the form \( M(x, y) \, dx + N(x, y) \, dy = 0 \), where \( M \) and \( N \) are continuous functions of \( x \) and \( y \).
In the given exercise, you encounter a first-order differential equation, indicated by the presence of \( dx \) and \( dy \) without derivatives of higher order. This type of equation forms the basis for analyzing many natural phenomena, including motion and growth.

Variable substitution is a powerful method used to simplify differential equations by introducing a new variable. In the context of homogeneous equations, we often use the substitution \( v = \frac{y}{x} \).
This substitution translates the dependent and independent variables \( x \) and \( y \) into a single variable \( v \), turning a two-variable problem into a simpler form. Let's imagine you're standing on the shoulders of this idea, allowing the differential equation to be easier to crack.

• With substitution, \( y = vx \).
• Also, differentiate to find \( dy = v \, dx + x \, dv \).

Using these relations transforms the original equation into one purely involving \( v \) and \( x \), thereby smoothing the path toward integration.

Separation of variables is a technique for solving differential equations where variables are rearranged to facilitate integration. This approach is particularly useful when the equation can be expressed so that each side contains only one of the original variables.
After applying the substitution in our example, the equation is reduced into a manageable form, allowing us to separate \( dx \) and \( dv \). Thus, transforming:

• \(\frac{dx}{x} = -\frac{v^2 - 1}{1 + v} \, dv \)

This separation aligns each side with a single variable, leading toward integration. It lays the groundwork for elucidating the intricate behaviors entailed by the differential equation.

Integration is the process of solving a differential equation by finding the antiderivatives. After separation of variables, this process helps us unite the components of the differential equation.
In the exercise, we integrate both sides after separation:

• \(\int \frac{1}{x} \, dx = \int -\frac{v^2 - 1}{1 + v} \, dv \)

This yields the solution in terms of natural logarithms and algebraic expressions, reflecting the accumulated effect over continuous movement.
Integration culminates in expressing the original problem within a comprehensible framework, revealing the general solution: \[ x \, e^{\frac{y^2}{2x^2}} = C(1 + \frac{y}{x}) \] This general solution encapsulates the essence of the differential equation, highlighting the beauty and application of calculus.