The given differential equation is:

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

This is a **non-linear** first-order differential equation, but we can solve it using the method of **separation of variables** by rewriting the equation in terms of \( y/x \).

Let’s introduce the substitution \( v = \frac{y}{x} \), which implies \( y = vx \) and \( \frac{dy}{dx} = v + x\frac{dv}{dx} \).

### Step 1: Substitution
Substitute \( y = vx \) into the original equation:

\[
\frac{dy}{dx} = \frac{x^2 - 5(vx)^2}{2x(vx)}
\]
Simplify the right-hand side:

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

### Step 2: Substituting for \(\frac{dy}{dx}\)
Now substitute \( \frac{dy}{dx} = v + x\frac{dv}{dx} \) into the equation:

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

### Step 3: Simplify and Separate Variables
Rearrange the equation to isolate \( \frac{dv}{dx} \) on one side:

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

Simplify the right-hand side:

\[
x\frac{dv}{dx} = \frac{1 - 5v^2 - 2v^2}{2v} = \frac{1 - 7v^2}{2v}
\]

### Step 4: Separate Variables
Now separate \( v \) and \( x \):

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

### Step 5: Integrate Both Sides
Integrate both sides:

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

The left-hand side can be integrated using the substitution \( u = 1 - 7v^2 \), and the right-hand side gives \( \ln|x| \).

Let me calculate the integral.

\[
u = 1 - 7v^2 \quad \Rightarrow \quad du = -14v dv
\]

Thus, we need to compute:

\[
\int \frac{2v}{1 - 7v^2} dv = -\frac{1}{7} \int \frac{du}{u} = -\frac{1}{7} \ln|u| = -\frac{1}{7} \ln|1 - 7v^2|
\]

So, after integrating both sides, we get:

\[
-\frac{1}{7} \ln|1 - 7v^2| = \ln|x| + C
\]

### Step 6: Simplify
Multiply both sides by -7:

\[
\ln|1 - 7v^2| = -7\ln|x| - 7C
\]

We can rewrite this as:

\[
\ln|1 - 7v^2| = \ln\left(\frac{1}{x^7}\right) + C'
\]

Exponentiate both sides to remove the logarithm:

\[
|1 - 7v^2| = \frac{C'}{x^7}
\]

### Step 7: Substitute \( v = \frac{y}{x} \)
Finally, substitute back \( v = \frac{y}{x} \) into the equation:

\[
|1 - 7\left(\frac{y}{x}\right)^2| = \frac{C'}{x^7}
\]

Thus, the general solution is:

\[
|1 - 7\frac{y^2}{x^2}| = \frac{C'}{x^7}
\]

Do you want further details or have any questions?

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Here are 5 related questions for practice:
1. Can you solve the differential equation \(\frac{dy}{dx} = \frac{x^2 + 3y^2}{xy}\)?
2. What substitution would work for \(\frac{dy}{dx} = \frac{x + y}{x - y}\)?
3. How do we solve \(\frac{dy}{dx} = \frac{y}{x}\) using separation of variables?
4. What happens if the equation was \(\frac{dy}{dx} = x^2y\)?
5. How do you interpret solutions of separable differential equations graphically?

Tip: When separating variables, always check if the function can be rewritten in terms of one variable easily.

Explore how to solve the non-linear differential equation dy/dx = (x^2 - 5y^2) / (2xy) using the method of separation of variables and substitution. This solution walks through the steps of transforming the equation into simpler terms, integrating both sides, and finding the general solution. Suitable for advanced high school or college-level students.

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