Solve the differential equation by separation of variables: $\frac{dy}{dx}=x(1-y^2)$. Which of the following is the general solution?

• A. $\ln \left|y\right|=\frac{1}{2}{x}^2+C$
• B. $y=e^{\frac{1}{2}{x}^2+C}$
• C. $\arctan \left(y\right)=\frac{1}{2}{x}^2+C$
• D. $\mathrm{arctanh}\left(y\right)=\frac{1}{2}{x}^2+C$

Answer: (D)

Verified step by step guidance

1. Step 1: Recognize that the differential equation dy/dx = x (1 - y^2) can be solved using the method of separation of variables. This method involves rearranging the equation so that all terms involving y are on one side and all terms involving x are on the other side.
2. Step 2: Rewrite the equation as (1/(1 - y^2)) dy = x dx. This separates the variables y and x, allowing us to integrate each side independently.
3. Step 3: Integrate both sides. For the left-hand side, ∫(1/(1 - y^2)) dy, recognize that this is the derivative of the inverse hyperbolic tangent function, arctanh(y). For the right-hand side, ∫x dx, the integral is (1/2)x^2.
4. Step 4: Combine the results of the integration. The left-hand side becomes arctanh(y), and the right-hand side becomes (1/2)x^2 + C, where C is the constant of integration.
5. Step 5: Write the general solution as arctanh(y) = (1/2)x^2 + C. This matches the correct answer provided in the problem.