Step 1: Understanding the Concept:
A differential equation is called homogeneous if the right-hand side can be expressed as a function of the ratio \(\frac{y}{x}\).
In this problem, every term in the numerator and denominator has the same total degree (degree 2), which confirms it is homogeneous.
Such equations are solved by a change of variables that reduces the equation into a separable format, which is easier to integrate.
Step 2: Key Formula or Approach:
The standard substitution for homogeneous equations is \(y = vx\).
Differentiating this substitution using the product rule gives:
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]
Step 3: Detailed Explanation:
Rewrite the original equation by splitting the fraction:
\[ \frac{dy}{dx} = \frac{x^2}{xy} + \frac{y^2}{xy} = \frac{x}{y} + \frac{y}{x} \]
Now apply the substitution \(y = vx \implies \frac{y}{x} = v\) and \(\frac{x}{y} = \frac{1}{v}\):
\[ v + x \frac{dv}{dx} = \frac{1}{v} + v \]
Subtract \(v\) from both sides of the equation:
\[ x \frac{dv}{dx} = \frac{1}{v} \]
Now, separate the variables by bringing all terms with \(v\) to one side and \(x\) to the other:
\[ v \, dv = \frac{1}{x} dx \]
Integrate both sides:
\[ \int v \, dv = \int \frac{1}{x} dx \]
\[ \frac{v^2}{2} = \ln|x| + C_1 \]
Multiply the entire equation by 2:
\[ v^2 = 2 \ln|x| + 2C_1 \]
Let \(2C_1 = C\) (a new constant of integration):
\[ v^2 = 2 \ln|x| + C \]
Now, substitute back the original variable \(v = \frac{y}{x}\):
\[ \left(\frac{y}{x}\right)^2 = 2 \ln|x| + C \]
\[ \frac{y^2}{x^2} = 2 \ln|x| + C \]
Finally, multiply by \(x^2\) to obtain the explicit general solution:
\[ y^2 = 2x^2 \ln|x| + Cx^2 \]
Step 4: Final Answer:
The general solution of the differential equation is \(y^2 = 2x^2 \ln|x| + Cx^2\).
This matches option (A).