1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} = y(xy^3 - 1)$$ using the substitution method. 2. **Rewrite the equation:** The equation is $$\frac{dy}{dx} = y(xy^3 - 1) = y^4 x - y$$. 3. **Substitution:** Let $$v = y^4$$. Then, $$y = v^{1/4}$$ and $$\frac{dy}{dx} = \frac{1}{4} v^{-3/4} \frac{dv}{dx}$$. 4. **Rewrite the original equation in terms of $$v$$:** $$\frac{dy}{dx} = y^4 x - y = v x - v^{1/4}$$. 5. **Substitute $$\frac{dy}{dx}$$:** $$\frac{1}{4} v^{-3/4} \frac{dv}{dx} = v x - v^{1/4}$$. 6. **Multiply both sides by $$4 v^{3/4}$$:** $$\frac{dv}{dx} = 4 v^{3/4} (v x - v^{1/4}) = 4 v^{3/4} v x - 4 v^{3/4} v^{1/4} = 4 v^{7/4} x - 4 v$$. 7. **Simplify powers:** $$\frac{dv}{dx} = 4 x v^{7/4} - 4 v$$. 8. **Rewrite as:** $$\frac{dv}{dx} + 4 v = 4 x v^{7/4}$$. 9. **This is a nonlinear differential equation in $$v$$.** To solve, separate variables if possible or use an integrating factor if linear. Here, the term $$v^{7/4}$$ prevents linearity. 10. **Try substitution:** Let $$w = v^{ -3/4}$$, then $$v = w^{-4/3}$$. 11. **Calculate $$\frac{dv}{dx}$$ in terms of $$w$$:** $$\frac{dv}{dx} = \frac{d}{dx} (w^{-4/3}) = -\frac{4}{3} w^{-7/3} \frac{dw}{dx}$$. 12. **Substitute into the equation:** $$-\frac{4}{3} w^{-7/3} \frac{dw}{dx} + 4 w^{-4/3} = 4 x (w^{-4/3})^{7/4}$$. 13. **Simplify powers:** $$ (w^{-4/3})^{7/4} = w^{-7/3}$$. 14. **Rewrite:** $$-\frac{4}{3} w^{-7/3} \frac{dw}{dx} + 4 w^{-4/3} = 4 x w^{-7/3}$$. 15. **Multiply both sides by $$w^{7/3}$$:** $$-\frac{4}{3} \frac{dw}{dx} + 4 w^{( -4/3 + 7/3)} = 4 x$$. 16. **Simplify exponent:** $$-\frac{4}{3} \frac{dw}{dx} + 4 w^{1} = 4 x$$. 17. **Rewrite:** $$-\frac{4}{3} \frac{dw}{dx} + 4 w = 4 x$$. 18. **Divide entire equation by 4:** $$-\frac{1}{3} \frac{dw}{dx} + w = x$$. 19. **Multiply both sides by 3:** $$-\frac{dw}{dx} + 3 w = 3 x$$. 20. **Rewrite:** $$\frac{dw}{dx} - 3 w = -3 x$$. 21. **This is a linear first-order ODE.** Use integrating factor $$\mu(x) = e^{-3x}$$. 22. **Multiply both sides by $$\mu(x)$$:** $$e^{-3x} \frac{dw}{dx} - 3 e^{-3x} w = -3 x e^{-3x}$$. 23. **Left side is derivative:** $$\frac{d}{dx} (w e^{-3x}) = -3 x e^{-3x}$$. 24. **Integrate both sides:** $$w e^{-3x} = \int -3 x e^{-3x} dx + C$$. 25. **Integrate by parts:** Let $$u = x$$, $$dv = -3 e^{-3x} dx$$. 26. **Calculate:** $$du = dx$$, $$v = e^{-3x}$$. 27. **Integral:** $$\int -3 x e^{-3x} dx = x e^{-3x} - \int e^{-3x} dx = x e^{-3x} + \frac{1}{3} e^{-3x} + C'$$. 28. **Substitute back:** $$w e^{-3x} = x e^{-3x} + \frac{1}{3} e^{-3x} + C$$. 29. **Multiply both sides by $$e^{3x}$$:** $$w = x + \frac{1}{3} + C e^{3x}$$. 30. **Recall substitution:** $$w = v^{-3/4} = y^{-3}$$. 31. **Final implicit solution:** $$y^{-3} = x + \frac{1}{3} + C e^{3x}$$. 32. **Solve for $$y$$ if needed:** $$y = \left(x + \frac{1}{3} + C e^{3x}\right)^{-1/3}$$. **Answer:** $$y = \left(x + \frac{1}{3} + C e^{3x}\right)^{-1/3}$$ where $$C$$ is an arbitrary constant.