1. **State the problem:** Solve the differential equation $$(1 + y^4) \frac{dy}{dx} = x^3 e^{x^2}$$ and identify the correct general solution from the given options. 2. **Rewrite the equation:** Separate variables to isolate $y$ terms on one side and $x$ terms on the other: $$ (1 + y^4) dy = x^3 e^{x^2} dx $$ 3. **Integrate both sides:** - Left side: $$\int (1 + y^4) dy = \int 1 dy + \int y^4 dy = y + \frac{y^5}{5} + C_1$$ - Right side: $$\int x^3 e^{x^2} dx$$ 4. **Evaluate the right integral:** Use substitution: Let $$u = x^2 \Rightarrow du = 2x dx \Rightarrow x dx = \frac{du}{2}$$ Rewrite the integral: $$\int x^3 e^{x^2} dx = \int x^2 \cdot x e^{x^2} dx = \int u \cdot e^u \cdot \frac{du}{2} = \frac{1}{2} \int u e^u du$$ 5. **Integrate by parts:** Let $$v = u$$ and $$dw = e^u du$$ Then $$dv = du$$ and $$w = e^u$$ Integration by parts formula: $$\int v dw = vw - \int w dv$$ Apply: $$\int u e^u du = u e^u - \int e^u du = u e^u - e^u + C = e^u (u - 1) + C$$ 6. **Substitute back:** $$\int x^3 e^{x^2} dx = \frac{1}{2} e^{x^2} (x^2 - 1) + C_2$$ 7. **Combine results:** $$ y + \frac{y^5}{5} = \frac{1}{2} e^{x^2} (x^2 - 1) + C $$ 8. **Compare with options:** This matches option A. **Final answer:** $$ y + \frac{y^5}{5} = \frac{1}{2} (x^2 - 1) e^{x^2} + C $$