Solve the differential equation: $\frac{dy}{dx}=3x^2{y}^2$. Which of the following is the general solution?

• A. $y=3x^2+C$
• B. $y=x^3+C$
• C. $y=e^{3x^2}+C$
• D. $y=-\frac{1}{x^3+C}$

Answer: (D)

Verified step by step guidance

1. Step 1: Recognize that the given differential equation is separable, meaning it can be written in the form f(y)dy = g(x)dx. Rewrite the equation as (1/y^2)dy = 3x^2dx.
2. Step 2: Integrate both sides of the equation. For the left-hand side, integrate ∫(1/y^2)dy, which simplifies to -1/y. For the right-hand side, integrate ∫3x^2dx, which simplifies to x^3 + C, where C is the constant of integration.
3. Step 3: Combine the results of the integration to form the equation -1/y = x^3 + C.
4. Step 4: Solve for y by isolating it. Multiply through by -1 to get y = -1/(x^3 + C).
5. Step 5: Verify that the solution matches the given general solution y = -1/(x^3 + C). This confirms that the correct answer is indeed the provided solution.