Question Number 14483 by chux last updated on 01/Jun/17
$$\mathrm{x}^{\mathrm{y}} +\mathrm{y}^{\mathrm{x}} =\mathrm{3}.....\left(\mathrm{1}\right) \\ $$$$\mathrm{x}+\mathrm{y}=\mathrm{3}.....\left(\mathrm{2}\right) \\ $$$$ \\ $$$$\mathrm{solve}\:\mathrm{the}\:\mathrm{equation} \\ $$
Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 01/Jun/17
$$\left({x},{y}\right)=\left(\mathrm{1},\mathrm{2}\right) \\ $$
Commented by chux last updated on 01/Jun/17
$$\mathrm{please}\:\mathrm{show}\:\mathrm{workings} \\ $$
Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 01/Jun/17
$${y}^{{x}} =\mathrm{3}−{x}^{{y}} \Rightarrow{x}^{{y}} <\mathrm{3}\Rightarrow{ylogx}<{log}\mathrm{3} \\ $$$${x}^{{y}} =\mathrm{3}−{y}^{{x}} \Rightarrow{y}^{{x}} <\mathrm{3}\Rightarrow{xlogy}<{log}\mathrm{3} \\ $$$${ylogx}−{xlogy}<\mathrm{0}\Rightarrow{x}^{{y}} <{y}^{{x}} \Rightarrow{x}<{y} \\ $$$${x}+{y}=\mathrm{3},{x}<{y}\Rightarrow{x}=\mathrm{1},{y}=\mathrm{2}\:. \\ $$
Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 01/Jun/17
Commented by chux last updated on 01/Jun/17
$$\mathrm{thanks}....\:\mathrm{please}\:\mathrm{can}\:\mathrm{it}\:\mathrm{be}\:\mathrm{solved} \\ $$$$\mathrm{by}\:\mathrm{any}\:\mathrm{numerical}\:\mathrm{method}.\: \\ $$$$ \\ $$
Commented by chux last updated on 01/Jun/17
$$\mathrm{please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{sirs} \\ $$
Commented by mrW1 last updated on 01/Jun/17
$${For}\:{equations}\:{like}\:{x}^{{y}} +{y}^{{x}} ={a}\:{there}\:{is} \\ $$$${no}\:{general}\:{analytical}\:{solution}.\:{For} \\ $$$${special}\:{values}\:{as}\:{in}\:{your}\:{case}\:{you} \\ $$$${can}\:{find}\:{the}\:{right}\:{solution} \\ $$$${through}\:``{try}\:{and}\:{error}'',\:``{guess}''\:{or} \\ $$$${intuition}.\:{But}\:{for}\:{any}\:{other}\:{values} \\ $$$${you}\:{can}\:{only}\:{get}\:{numerical}\:{approximation} \\ $$$${of}\:{the}\:{solution}.\:{An}\:{usual}\:{way}\:{is} \\ $$$${using}\:{graphics}. \\ $$$${For}\:{example} \\ $$$${x}^{{y}} +{y}^{{x}} =\mathrm{3}.\mathrm{5} \\ $$$${x}+{y}=\mathrm{5}.\mathrm{5} \\ $$$${you}\:{can}\:{get}\:\mathrm{2}\:{solutions}\:{using}\:{graphic} \\ $$$${x}\approx\mathrm{0}.\mathrm{7544}\:{or}\:\mathrm{4}.\mathrm{7456} \\ $$$${y}\approx\mathrm{4}.\mathrm{7456}\:{or}\:\mathrm{0}.\mathrm{7544} \\ $$$${Here}\:{you}\:{can}\:{not}\:{guess}\:{the}\:{solution}. \\ $$
Commented by chux last updated on 01/Jun/17
$$\mathrm{Thanks}\:.....\:\mathrm{I}\:\mathrm{really}\:\mathrm{appreciate}\:\mathrm{this}. \\ $$