1. **State the problem:** Solve the partial differential equation (PDE) given by $$\frac{dy}{dx} = \frac{3y^2 - x^2}{2xy}$$. 2. **Rewrite the equation:** The equation can be written as $$\frac{dy}{dx} = \frac{3y^2}{2xy} - \frac{x^2}{2xy} = \frac{3y}{2x} - \frac{x}{2y}$$. 3. **Separate variables:** Multiply both sides by $2xy$ to clear the denominator: $$2xy \frac{dy}{dx} = 3y^2 - x^2$$ 4. **Rewrite as:** $$2xy \frac{dy}{dx} + x^2 = 3y^2$$ 5. **Divide both sides by $x^2$ to simplify:** $$\frac{2xy}{x^2} \frac{dy}{dx} + \frac{x^2}{x^2} = \frac{3y^2}{x^2}$$ 6. **Simplify fractions:** $$2 \frac{y}{x} \frac{dy}{dx} + 1 = 3 \left(\frac{y}{x}\right)^2$$ 7. **Let $v = \frac{y}{x}$, then $y = vx$ and $\frac{dy}{dx} = v + x \frac{dv}{dx}$ (by product rule).** 8. **Substitute into the equation:** $$2v (v + x \frac{dv}{dx}) + 1 = 3v^2$$ 9. **Expand:** $$2v^2 + 2vx \frac{dv}{dx} + 1 = 3v^2$$ 10. **Rearrange terms:** $$2vx \frac{dv}{dx} = 3v^2 - 2v^2 - 1 = v^2 - 1$$ 11. **Divide both sides by $2v x$:** $$\frac{dv}{dx} = \frac{v^2 - 1}{2v x}$$ 12. **Separate variables:** $$\frac{2v}{v^2 - 1} dv = \frac{1}{x} dx$$ 13. **Integrate both sides:** $$\int \frac{2v}{v^2 - 1} dv = \int \frac{1}{x} dx$$ 14. **Use substitution for left integral:** Let $u = v^2 - 1$, then $du = 2v dv$, so $$\int \frac{2v}{v^2 - 1} dv = \int \frac{1}{u} du = \ln|u| + C = \ln|v^2 - 1| + C$$ 15. **Integrate right side:** $$\int \frac{1}{x} dx = \ln|x| + C$$ 16. **Equate integrals:** $$\ln|v^2 - 1| = \ln|x| + C$$ 17. **Exponentiate both sides:** $$|v^2 - 1| = C_1 |x|$$ 18. **Recall $v = \frac{y}{x}$:** $$\left|\left(\frac{y}{x}\right)^2 - 1\right| = C_1 |x|$$ 19. **Multiply both sides by $x^2$:** $$|y^2 - x^2| = C_1 |x|^3$$ 20. **General solution:** $$y^2 - x^2 = C x^3$$ where $C$ is an arbitrary constant. **Final answer:** $$y^2 - x^2 = C x^3$$