Let y = y(x) be the solution curve of the differential equation (1 + x) dy dx + (y+1) x = 0, y(0) = 0. If the curve y = y(x) passes through the point ( , -1 2 ), then a value of is :

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Correct answer: D. 2

The given differential equation can be written as: (1 + x)dy + (y+1) x dx = 0 d((y+1)(1 + x)) = 0 Integrating both sides: (y+1)(1 + x) = C Given y(0) = 0, substituting x = 0 and y = 0: (0+1)(1 + 0) = C C = 1 The equation of the curve is (y+1)(1 + x) = 1 Since the curve passes through ( , -1 2 ), substituting x = and y = -1 2 : ( -1 2 + 1 )(1 + ) = 1 1 2 (1 + ) = 1 1 + = 2 = 1 = 2 Answer: 2

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