1. The problem is to solve the first differential equation given: $$\frac{dy}{dx} + x^2 y^2 = x$$. 2. This is a first-order nonlinear ordinary differential equation because of the $y^2$ term. 3. We can try to solve it by rearranging terms: $$\frac{dy}{dx} = x - x^2 y^2$$ 4. Rewrite as: $$\frac{dy}{dx} = x - x^2 y^2$$ 5. This is not separable in its current form, but we can try substitution or other methods. Let's try substitution $v = y^{-1}$, so $y = \frac{1}{v}$. 6. Then, $$\frac{dy}{dx} = -\frac{1}{v^2} \frac{dv}{dx}$$. 7. Substitute into the equation: $$-\frac{1}{v^2} \frac{dv}{dx} + x^2 \frac{1}{v^2} = x$$ Multiply both sides by $v^2$: $$-\frac{dv}{dx} + x^2 = x v^2$$ 8. Rearranged: $$-\frac{dv}{dx} = x v^2 - x^2$$ or $$\frac{dv}{dx} = x^2 - x v^2$$ 9. This is still nonlinear, but let's write it as: $$\frac{dv}{dx} = x^2 - x v^2$$ 10. This is a Bernoulli equation in $v$: $$\frac{dv}{dx} + x v^2 = x^2$$ 11. Bernoulli equations have the form: $$\frac{dy}{dx} + P(x) y = Q(x) y^n$$ Here, $n=2$, $P(x) = 0$, but the equation is slightly different. Let's try substitution $w = v^{-1}$. 12. Then, $$\frac{dw}{dx} = -v^{-2} \frac{dv}{dx}$$. 13. From step 10, multiply both sides by $-v^{-2}$: $$-v^{-2} \frac{dv}{dx} - x = -x^2 v^{-2}$$ But this is complicated; instead, let's solve the original equation numerically or by other methods. 14. Since the problem is complex, the key takeaway is that the first equation is nonlinear and requires advanced methods or numerical solutions. Final answer: The first differential equation $$\frac{dy}{dx} + x^2 y^2 = x$$ is nonlinear and does not have a straightforward closed-form solution using elementary methods.