Question Number 17322 by tawa tawa last updated on 04/Jul/17
$$\mathrm{Solve}:\:\:\:\frac{\mathrm{dy}}{\mathrm{dx}}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{y}\:=\:\frac{\mathrm{3}}{\mathrm{2}}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{with}\:\:\:\mathrm{y}\left(\mathrm{0}\right)\:=\:\mathrm{4} \\ $$
Answered by mrW1 last updated on 04/Jul/17
$$\mathrm{2}\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{3}−\mathrm{y} \\ $$$$−\frac{\mathrm{dy}}{\mathrm{y}−\mathrm{3}}=\frac{\mathrm{dx}}{\mathrm{2}} \\ $$$$\int\frac{\mathrm{dy}}{\mathrm{y}−\mathrm{3}}=−\int\frac{\mathrm{dx}}{\mathrm{2}} \\ $$$$\mathrm{ln}\:\mid\mathrm{y}−\mathrm{3}\mid=−\frac{\mathrm{x}}{\mathrm{2}}+\mathrm{C} \\ $$$$\mathrm{y}\left(\mathrm{0}\right)=\mathrm{4} \\ $$$$\mathrm{ln}\:\left(\mathrm{1}\right)=\mathrm{0}+\mathrm{C} \\ $$$$\Rightarrow\mathrm{C}=\mathrm{0} \\ $$$$\mathrm{ln}\:\mid\mathrm{y}−\mathrm{3}\mid=−\frac{\mathrm{x}}{\mathrm{2}} \\ $$$$\mathrm{y}−\mathrm{3}=\pm\mathrm{e}^{−\frac{\mathrm{x}}{\mathrm{2}}} \\ $$$$\mathrm{y}=\mathrm{3}\pm\mathrm{e}^{−\frac{\mathrm{x}}{\mathrm{2}}} \\ $$$$ \\ $$$$\mathrm{since}\:\mathrm{y}\left(\mathrm{0}\right)=\mathrm{4} \\ $$$$\Rightarrow\:\mathrm{y}=\mathrm{3}+\mathrm{e}^{−\frac{\mathrm{x}}{\mathrm{2}}} \\ $$
Commented by tawa tawa last updated on 04/Jul/17
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$
Commented by tawa tawa last updated on 04/Jul/17
$$\mathrm{How}\:\mathrm{is}\:\mathrm{C}\:=\:\mathrm{0}\:\mathrm{sir}\:??? \\ $$
Commented by mrW1 last updated on 04/Jul/17
$$\mathrm{i}\:\mathrm{changed}\:\mathrm{my}\:\mathrm{answer}.\:\mathrm{there}\:\mathrm{are}\:\mathrm{2} \\ $$$$\mathrm{possible}\:\mathrm{solutions}. \\ $$
Commented by tawa tawa last updated on 04/Jul/17
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$$$ \\ $$