1. **Problem:** Solve the first-order ODE $$\frac{dy}{dx} = -\frac{6x^2 y^2 + y^5}{6x^3 y^2 + 5xy^4}$$ 2. **Step 1: Rewrite the equation** We can write the ODE as $$\frac{dy}{dx} = -\frac{6x^2 y^2 + y^5}{6x^3 y^2 + 5xy^4}$$ 3. **Step 2: Check if the equation is separable** Try to express as $\frac{dy}{dx} = g(x)h(y)$ or separate variables. 4. **Step 3: Check if the equation is exact** Rewrite as $$\left(6x^3 y^2 + 5xy^4\right) dy + \left(6x^2 y^2 + y^5\right) dx = 0$$ Let $$M = 6x^2 y^2 + y^5, \quad N = 6x^3 y^2 + 5xy^4$$ Check exactness: $$\frac{\partial M}{\partial y} = 12x^2 y + 5y^4$$ $$\frac{\partial N}{\partial x} = 18x^2 y^2 + 5y^4$$ Since $\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$, the equation is not exact. 5. **Step 4: Check if it is homogeneous** Both numerator and denominator are homogeneous functions of degree 5: - Numerator: $6x^2 y^2$ degree $2+2=4$ plus $y^5$ degree 5, so not homogeneous of same degree. - Denominator: $6x^3 y^2$ degree $3+2=5$, $5xy^4$ degree $1+4=5$. Since numerator terms have different degrees, not homogeneous. 6. **Step 5: Try substitution $v = y/x$** Rewrite $y = vx$, then $dy/dx = v + x dv/dx$. Substitute into the ODE: $$v + x \frac{dv}{dx} = -\frac{6x^2 (v x)^2 + (v x)^5}{6x^3 (v x)^2 + 5x (v x)^4} = -\frac{6x^2 v^2 x^2 + v^5 x^5}{6x^3 v^2 x^2 + 5x v^4 x^4} = -\frac{6 v^2 x^4 + v^5 x^5}{6 v^2 x^5 + 5 v^4 x^5}$$ Simplify powers of $x$: $$= -\frac{x^4 (6 v^2 + v^5 x)}{x^5 (6 v^2 + 5 v^4)} = -\frac{6 v^2 + v^5 x}{x (6 v^2 + 5 v^4)}$$ This is complicated; try to factor differently. 7. **Step 6: Simplify numerator and denominator by $x^4$** Rewrite numerator: $$6 v^2 + v^5 x = 6 v^2 + v^5 x$$ Denominator: $$x (6 v^2 + 5 v^4)$$ Since $v = y/x$, $x$ is independent variable, so this substitution is not simplifying well. 8. **Step 7: Alternative approach: Multiply numerator and denominator by $1/(x y^2)$** Rewrite ODE as $$\frac{dy}{dx} = -\frac{6x^2 y^2 + y^5}{6x^3 y^2 + 5xy^4} = -\frac{6x^2 y^2 / (x y^2) + y^5 / (x y^2)}{6x^3 y^2 / (x y^2) + 5xy^4 / (x y^2)} = -\frac{6x + y^3 / x}{6x^2 + 5 y^2}$$ 9. **Step 8: Let $z = y^3 / x$** Try to express in terms of $z$ and $x$ to simplify. 10. **Step 9: Due to complexity, the best method is to solve implicitly or numerically.** **Final answer:** The ODE is not separable, not exact, and not homogeneous in a straightforward way. Substitution attempts do not simplify it easily. The solution requires implicit or numerical methods beyond the scope here. --- **Summary:** The first ODE is complex and does not fit standard methods easily. Further advanced techniques or numerical methods are recommended.