Let $y=y(x)$ be the solution curve of the differential equation $\frac{d y}{d x}+\frac{1}{x^{2}-1} y=\left(\frac{x-1}{x+1}\right)^{1 / 2}$, $x >1$ passing through the point $\left(2, \sqrt{\frac{1}{3}}\right)$. Then $\sqrt{7}\, y(8)$ is equal to :

<p>$${{dy} \over {dx}} + {1 \over {{x^2} - 1}}y = \sqrt {{{x - 1} \over {x + 1}}} ,\,x > 1$$</p> <p>Integrating factor I.F. $$ = {e^{\int {{1 \over {{x^2} - 1}}dx} }} = {e^{{1 \over 2}\ln \left| {{{x - 1} \over {x + 1}}} \right|}}$$</p> <p>$= \sqrt {{{x - 1} \over {x + 1}}}$</p> <p>Solution of differential equation</p> <p>$$y\sqrt {{{x - 1} \over {x + 1}}} = \int {{{x - 1} \over {x + 1}}dx = \int {\left( {1 - {2 \over {x + 1}}} \right)dx} } $$</p> <p>$y\sqrt {{{x - 1} \over {x + 1}}} = x - 2\ln |x + 1| + C$</p> <p>Curve passes through $\left( {2,\sqrt {{1 \over 3}} } \right)$</p> <p>${1 \over {\sqrt 3 }} \times {1 \over {\sqrt 3 }} = 2 - 2\ln 3 + C$</p> <p>$C = 2\ln 3 - {5 \over 3}$</p> <p>$y(8) \times {{\sqrt 7 } \over 3} = 8 - 2\ln 9 + 2\ln 3 - {5 \over 3}$</p> <p>$\sqrt 7 \,.\,y(8) = 19 - 6\ln 3$</p>