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