Question Number 68925 by ramirez105 last updated on 16/Sep/19
Commented by kaivan.ahmadi last updated on 16/Sep/19
$$\frac{\partial{u}}{\partial{x}}=\left({x}+\mathrm{3}\right)^{−\mathrm{1}} {cosy}\Rightarrow{u}\left({x},{y}\right)={cosyln}\left({x}+\mathrm{3}\right)+{h}\left({y}\right) \\ $$$$\frac{\partial{u}}{\partial{y}}={y}^{−\mathrm{1}} −{sinyln}\left(\mathrm{5}{x}+\mathrm{15}\right)=−{sinyln}\left({x}+\mathrm{3}\right)+\frac{{dh}}{{dy}}\Rightarrow \\ $$$${y}^{−\mathrm{1}} −{ln}\mathrm{5}=\frac{{dh}}{{dy}}\Rightarrow{h}={lny}−{yln}\mathrm{5} \\ $$$${u}\left({x},{y}\right)={c}\Rightarrow{cosyln}\left({x}+\mathrm{3}\right)+{lny}−{yln}\mathrm{5}={c} \\ $$
Commented by ramirez105 last updated on 17/Sep/19
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir}! \\ $$