To solve the given differential equation, we have:
[tex]\[
\frac{d y}{d x} = -\frac{2 x y - 3 x^2}{x^2 - 2 y}
\][/tex]
Let's break down the process step-by-step:
1. Identify the form of the differential equation:
We observe that the differential equation is given in the form:
[tex]\[
\frac{d y}{d x} = f(x, y)
\][/tex]
Here, the function [tex]\(f(x, y)\)[/tex] is:
[tex]\[
f(x, y) = -\frac{2 x y - 3 x^2}{x^2 - 2 y}
\][/tex]
2. Simplify the function [tex]\(f(x, y)\)[/tex]:
Our goal is to manipulate [tex]\(f(x, y)\)[/tex] to understand it better. By simplifying the negation in the numerator of the given expression, we get:
[tex]\[
-\frac{2 x y - 3 x^2}{x^2 - 2 y} = \frac{3 x^2 - 2 x y}{x^2 - 2 y}
\][/tex]
3. Interpret the simplified form:
At this stage, we have determined that:
[tex]\[
\frac{d y}{d x} = \frac{3 x^2 - 2 x y}{x^2 - 2 y}
\][/tex]
There may be multiple approaches to solve the differential equation further, such as separation of variables, integrating factors, or recognizing a particular solution type if it fits known forms. However, the crucial detail is identifying the correct form of [tex]\( \frac{d y}{d x} \)[/tex], which we have confirmed to be:
[tex]\[
\frac{3 x^2 - 2 x y}{x^2 - 2 y}
\][/tex]
This is the detailed solution to the differential equation provided.
