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 _____________.

Question

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 _____________.

Correct Answer

320

Solution

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