The given differential equation is:

\[
\frac{dy}{dx} = \frac{x^2 - 5y^2}{2xy}
\]

This is a homogeneous differential equation, which means we can solve it by using the substitution:

\[
v = \frac{y}{x} \quad \text{or} \quad y = vx \quad \Rightarrow \quad \frac{dy}{dx} = v + x\frac{dv}{dx}
\]

### Step 1: Substitute \( y = vx \)

By substituting \( y = vx \) into the equation \( \frac{dy}{dx} \), we get:

\[
v + x\frac{dv}{dx} = \frac{x^2 - 5(vx)^2}{2x(vx)}
\]

Simplifying the right-hand side:

\[
v + x\frac{dv}{dx} = \frac{x^2 - 5v^2x^2}{2vx} = \frac{x^2(1 - 5v^2)}{2vx}
\]

\[
v + x\frac{dv}{dx} = \frac{x(1 - 5v^2)}{2v}
\]

### Step 2: Simplify the equation

Cancel out \( x \) on both sides:

\[
v + x\frac{dv}{dx} = \frac{1 - 5v^2}{2v}
\]

Now, rearrange the equation:

\[
x\frac{dv}{dx} = \frac{1 - 5v^2}{2v} - v
\]

Combine the terms on the right-hand side:

\[
x\frac{dv}{dx} = \frac{1 - 5v^2 - 2v^2}{2v}
\]

\[
x\frac{dv}{dx} = \frac{1 - 7v^2}{2v}
\]

### Step 3: Separate variables

We can now separate the variables \( v \) and \( x \):

\[
\frac{2v}{1 - 7v^2} dv = \frac{dx}{x}
\]

### Step 4: Integrate both sides

Now, integrate both sides.

For the left-hand side, use the substitution \( u = 1 - 7v^2 \), hence \( du = -14v dv \), and adjust accordingly.

For the right-hand side, the integral is straightforward.

Would you like me to continue with the step-by-step integration and final solution, or provide details about any specific part?

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**5 Related Questions:**
1. How do you handle non-homogeneous differential equations?
2. What other types of substitutions can be used for solving differential equations?
3. Can this equation be solved using numerical methods instead of analytical ones?
4. What are the conditions for a differential equation to be homogeneous?
5. How can this method be applied to partial differential equations?

**Tip:** Always check if a differential equation is separable before trying other methods, as separable equations are generally easier to solve.

Discover how to solve the homogeneous differential equation dy/dx = (x^2 - 5y^2)/2xy using variable substitution and separation of variables. Follow a detailed, step-by-step approach suitable for calculus students studying differential equations.

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