Let y = y(x) be the solution of the differential equation (1 - x²)dy = (xy + (x³ + 2)√(1 - x²))dx, - 1 < x < 1, and y(0) = 0. If ∫ (- 1)/2 ^(1/2) √(1 - x²) y(x)dx = k, then k⁻¹ is equal to _ _ _ _ _ _ _.

Differential Equations: JEE Main 2022 Mathematics question

JEE Main 2022 · 27 Jun, Evening shift · Q33

JEE Main 202227 Jun, Evening shiftMathematicsMediumDifferential Equations163× since 2002

Let

y = y(x)

be the solution of the differential equation

(1 - {x^2})dy = \left( {xy + ({x^3} + 2)\sqrt {1 - {x^2}} } \right)dx, - 1 < x < 1

, and

y(0) = 0

. If

\int_{{{ - 1} \over 2}}^{{1 \over 2}} {\sqrt {1 - {x^2}} y(x)dx = k}

, then k^

-

1 is equal to _____________.

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