Question Number 184066 by Matica last updated on 02/Jan/23
$$\frac{{dy}}{{dx}}={y}\left({y}+\mathrm{2}\right).\:{find}\:\:{y}=? \\ $$
Answered by mr W last updated on 02/Jan/23
$$\int\frac{{dy}}{{y}\left({y}+\mathrm{2}\right)}=\int{dx} \\ $$$$\int\left(\frac{\mathrm{1}}{{y}}−\frac{\mathrm{1}}{{y}+\mathrm{2}}\right){dy}=\mathrm{2}\int{dx} \\ $$$$\mathrm{ln}\:{y}−\mathrm{ln}\:\left({y}+\mathrm{2}\right)=\mathrm{2}{x}+{C}_{\mathrm{1}} \\ $$$$\mathrm{ln}\:\frac{{y}}{{y}+\mathrm{2}}=\mathrm{2}{x}+{C}_{\mathrm{1}} \\ $$$$\frac{{y}}{{y}+\mathrm{2}}={Ce}^{\mathrm{2}{x}} \\ $$$$\Rightarrow{y}=\frac{\mathrm{2}}{\mathrm{1}−{Ce}^{\mathrm{2}{x}} }−\mathrm{2} \\ $$
Commented by Matica last updated on 02/Jan/23
$${Ok}.\:{Thank}\:{you}. \\ $$