Question

The general solution of $ \left( \frac{dy}{dx} \right)^2 = 1 - x^2 - y^2 + x^2y^2 $ is

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

Hint:

When dealing with differential equations that involve trigonometric functions, look for trigonometric identities to simplify the equation. In many cases, using \( \sin^{-1} \) or \( \cos^{-1} \) helps in solving such equations.

Answer 1

The Correct Option is A

Given the equation: \[ \left( \frac{dy}{dx} \right)^2 = 1 - x^2 - y^2 + x^2y^2 \] To solve this, we first separate variables and solve for \( y \) in terms of \( x \). The resulting expression involves trigonometric identities, which we use to integrate both sides of the equation. The general solution can be found as: \[ 2\sin^{-1}y = x\sqrt{1 - x^2} + \sin^{-1}x + C \] This represents the general solution to the differential equation.