91Ó°ÊÓ

One of the most useful techniques for solving differential equations is the method called "Separation of Variables." This method can be applied when the differential equation can be rewritten so that all the terms involving one variable (say, all terms with "y") are on one side of the equation, and all terms with the other variable ("x") are on the other side.
For the given differential equation \( \frac{1}{x} \frac{dy}{dx} = ye^{x^2} + 2\sqrt{y} e^{x^2} \), this technique is applicable. After manipulation, which includes multiplying by \(x\) to eliminate the fraction, the equation can be reorganized to get \( \frac{1}{y + 2\sqrt{y}} \, dy = xe^{x^2} \, dx \).
This step shows the separation as it allows each side of the equation to be integrated with respect to its own variable. This is a crucial simplification step, as separating the variables paves the way to integrating both sides, thus finding a solution to the equation. This would not be possible unless the equation is separated into distinct parts like this.

Integration is the reverse process of differentiation and is a key operation in solving differential equations through the separation of variables. To solve the differential equation after separating the variables, integrate both sides concerning their respective variables.
For the left side \( \int \frac{1}{y + 2\sqrt{y}} \, dy \), a substitution can simplify the integration process. By letting \( u = \sqrt{y} \), we transform it into a more accessible integral: \( \int \frac{2}{u+2} \, du \). This simplification makes the integration straightforward.
On the other hand, the right side \( \int xe^{x^2} \, dx \) requires another substitution, \( z = x^2 \), leading to \( \int \frac{1}{2} e^z \, dz \). Working through these integrations builds the foundation for expressing \( y \) in terms of \( x \). These techniques highlight the importance of integration methods like substitution in solving differential equations.

A first-order differential equation involves the first derivative of a function but no higher derivatives. It typically takes a form \( \frac{dy}{dx} = f(x, y) \) where \( f(x, y) \) can be a function involving both \(x\) and \(y\).
The differential equation given, \( \frac{1}{x} \frac{dy}{dx} = ye^{x^2} + 2\sqrt{y}e^{x^2} \), exemplifies a first-order equation because it only uses \( \frac{dy}{dx} \) (the first derivative of \( y \) with respect to \( x \)). The goal is to solve for \( y \) as a function of \( x \).
In such equations, one often looks to methods like separation of variables or other techniques to isolate and integrate the derivative terms. Solutions to first-order differential equations provide valuable insights into various real-world dynamics where systems change with respect to time or another independent variable.