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dy/dx + x y/(x^2 - 1) = polynomial/sqrt(1 - x^2) through the origin: integrate the solution

The same question, asked 3 times: 27 Jun 2022*, 26 Jul 2022*, 22 Jan 2025. Every wording below, oldest first.

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  1. Question 1

    27 Jun 2022

    178
    Let y=y(x)y = y(x) be the solution of the differential equation (1−x2)dy=(xy+(x3+2)1−x2)dx,−1<x<1\left( 1 -x^{2} \right)dy=\left( xy+\left( x^{3}+ 2 \right)\sqrt{1 -x^{2}} \right)dx, - 1 <x< 1and y(0)=0y(0) = 0 if ∫−12121−x2y(x)dx=k\int_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1 -x^{2}}y(x)dx=k then k−1k^{- 1} is equal to:

  2. Question 2

    26 Jul 2022

    160
    Let the solution curve y=f(x)y = f(x) of the differential equation dydx+xyx2−1=x4+2x1−x2,x∈(−1,1)\frac{dy}{dx}+\frac{xy}{x^{2}- 1}=\frac{x^{4}+ 2x}{\sqrt{1 -x^{2}}},x \in ( - 1,1) pass through the origin. Then ∫−3232f(x)dx\int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x)dx is equal to

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  3. Question 3

    22 Jan 2025

    146
    Let y=f(x)y=f(x) be the solution of the differential equation dydx+xyx2−1=x6+4x1−x2, −1<x<1\frac{dy}{dx}+\frac{xy}{x^{2}-1}=\frac{x^{6}+4x}{\sqrt{1-x^{2}}},\ -1<x<1, such that f(0)=0f(0)=0. If 6∫−1/21/2f(x)dx=2π−α6\int_{-1/2}^{1/2} f(x)dx = 2\pi-\alpha, then α2\alpha^{2} is equal to