Question Number 180550 by a.lgnaoui last updated on 13/Nov/22
$${Resoudre}\: \\ $$$${af}^{'} \left({x}\right)+\frac{{b}}{{f}\left({x}\right)}+{c}=\mathrm{0}\:\:\:\:\left({a},{b},{c}\right)\in\mathbb{R}^{\mathrm{3}} \\ $$
Answered by mr W last updated on 13/Nov/22
$${a}\frac{{dy}}{{dx}}=−\frac{{b}+{cy}}{{y}} \\ $$$$\frac{{aydy}}{{b}+{cy}}=−{dx} \\ $$$$\left(\mathrm{1}−\frac{{b}}{{cy}+{b}}\right){dy}=−\frac{{c}}{{a}}{dx} \\ $$$$\int\left(\mathrm{1}−\frac{{b}}{{cy}+{b}}\right){dy}=−\frac{{c}}{{a}}\int{dx} \\ $$$${y}−\frac{{b}}{{c}}\mathrm{ln}\:\mid{cy}+{b}\mid=−\frac{{cx}}{{a}}+{k} \\ $$$$\Rightarrow{y}+\frac{{cx}}{{a}}−\frac{{b}}{{c}}\mathrm{ln}\:\mid{cy}+{b}\mid={k} \\ $$
Commented by a.lgnaoui last updated on 14/Nov/22
$${thanks}\: \\ $$