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To solve the differential equation given by:

\[
y' = \frac{x^2 + y^2}{xy}
\]

we can proceed by simplifying and analyzing the equation. This form suggests a substitution approach or a method that leverages homogeneous equations.

### Step 1: Rewrite \( y' \) in terms of \(\frac{dy}{dx}\)
The equation tells us that:
\[
\frac{dy}{dx} = \frac{x^2 + y^2}{xy}
\]

### Step 2: Simplify the right-hand side
Rewrite the right-hand side by separating terms:
\[
\frac{dy}{dx} = \frac{x^2}{xy} + \frac{y^2}{xy} = \frac{x}{y} + \frac{y}{x}
\]

### Step 3: Use substitution
To simplify further, let’s use the substitution \( v = \frac{y}{x} \). Then, \( y = vx \), and differentiating both sides with respect to \( x \) gives:
\[
\frac{dy}{dx} = v + x\frac{dv}{dx}
\]

### Step 4: Substitute into the equation
Substituting \(\frac{dy}{dx}\) and \(\frac{y}{x} = v\) back into the original differential equation, we get:
\[
v + x\frac{dv}{dx} = \frac{x}{vx} + \frac{v}{x}
\]


Solve the differential equation problem: y' = (x^2 + y^2) / (xy)

A step-by-step solution to solve the differential equation y' = (x^2 + y^2) / (xy). This solution uses substitution and explores homogeneous equations, suitable for college-level mathematics.

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