Question

If $y=f(x)$ satisfies the differential equation $\left(1+x^2\right)f^{\prime }(x)=x(1-f(x)),f(0)=\frac{4}{3}$, then find the value of $f(\sqrt{8})+\frac{8}{9}$

Answer 1

Answer: 2

$f^{\prime }(x)+\frac{x}{1+x^2}\cdot f(x)=\frac{x}{1+x^2}$
$\text{ I.F }=\sqrt{1+x^2}$
$\therefore f(x)=1+\frac{1}{3\sqrt{1+x^2}}$
$\therefore f(\sqrt{8})+\frac{8}{9}=2$