Question

The general solution of the differential equation \( (1 - x^2) \frac{dy{dx} + 2xy = x(1 - x^2)^{\frac{1}{2}} \) is}

• A. \( y = \sqrt{1 - x^2} + c(1 - x^2) \)
• B. \( y = 2\sqrt{1 - x^2} + c \)
• C. \( y = 2\sqrt{1 - x^2} + c(1 + x^2) \)
• D. \( y\sqrt{1 - x^2} = c(1 - x^2) \)

Hint:

For solving separable differential equations, isolate \( dy/dx \) and integrate both sides. Use integration constants as needed for the solution.

Answer 1

The Correct Option is A

Step 1: Solve the differential equation.
The given differential equation is separable. We can write it in the form: \[ \frac{dy}{dx} = \frac{x(1 - x^2)^{\frac{1}{2}} - 2xy}{1 - x^2} \] After solving, the general solution of the differential equation is: \[ y = \sqrt{1 - x^2} + c(1 - x^2) \] Step 2: Conclusion.
The correct answer is (A) \( y = \sqrt{1 - x^2} + c(1 - x^2) \).