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Question Number 128667 by sarahvalencia last updated on 09/Jan/21
Commented by liberty last updated on 09/Jan/21
$$\:\left(\mathrm{1}\right)\:\mathrm{a}^{\mathrm{2}} \mathrm{da}\:+\:\mathrm{2ab}\:\mathrm{db}\:+\:\mathrm{b}^{\mathrm{2}} \:\mathrm{da}\:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{d}\left(\mathrm{ab}^{\mathrm{2}} \right)\:+\:\mathrm{a}^{\mathrm{2}} \:\mathrm{da}\:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\int\:\mathrm{d}\left(\mathrm{ab}^{\mathrm{2}} \right)\:+\:\int\:\mathrm{a}^{\mathrm{2}} \:\mathrm{da}\:=\:\int\:\mathrm{0}\:\mathrm{da} \\ $$$$\:\:\:\:\:\:\mathrm{ab}^{\mathrm{2}} \:+\:\frac{\mathrm{1}}{\mathrm{3}}\mathrm{a}^{\mathrm{3}} \:=\:\mathrm{C}\: \\ $$
Answered by mohammad17 last updated on 09/Jan/21
$$\left.\mathrm{1}\right){M}={a}^{\mathrm{2}} +{b}^{\mathrm{2}} \rightarrow{M}_{{b}} =\mathrm{2}{b}\:,{N}=\mathrm{2}{ab}\rightarrow{N}_{{a}} =\mathrm{2}{b} \\ $$$$ \\ $$$${M}_{{b}} ={N}_{{a}} =\mathrm{2}{b}\Rightarrow{The}\:{equation}\:{is}\:{exact} \\ $$$$ \\ $$$$\int{Mda}+\int{Ndb}={C} \\ $$$$ \\ $$$$\int\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right){da}+\int\mathrm{2}{abdb}={C} \\ $$$$ \\ $$$$\frac{{a}^{\mathrm{3}} }{\mathrm{3}}+\mathrm{2}{ab}^{\mathrm{2}} ={C} \\ $$$$\circleddash{mohammad} \\ $$
Commented by liberty last updated on 09/Jan/21
$$\:\mathrm{d}\left(\frac{\mathrm{a}^{\mathrm{3}} }{\mathrm{3}}+\mathrm{2ab}^{\mathrm{2}} \right)\:=\:\mathrm{0} \\ $$$$\:\mathrm{a}^{\mathrm{2}} \:\mathrm{da}\:+\:\mathrm{4ab}\:\mathrm{db}+\mathrm{2b}^{\mathrm{2}} \:\mathrm{da}\:=\:\mathrm{0}\:\: \\ $$$$\:\mathrm{wrong}\:. \\ $$
Answered by mohammad17 last updated on 09/Jan/21
$$\left.\mathrm{2}\right){by}\:{sepraple}\: \\ $$$$ \\ $$$$\int{dx}+\int\frac{{n}}{\mathrm{5}{n}^{\mathrm{2}} −{y}}{dy}={C} \\ $$$$ \\ $$$${x}−{nln}\mid\mathrm{5}{n}^{\mathrm{2}} −{y}\mid={C} \\ $$$$ \\ $$$$\circledast{mohammad} \\ $$