$\mathrm{eqns}:=\left\{x^a{y}^b-U\,1\+\frac{\mathrm{\μ}{x}^{a}a{y}^b}{x}\,1\+\frac{\mathrm{\μ}{x}^a{y}^{b}b}{y}\right\}$

$\mathrm{eqns}≔\left\{1+\frac{\mathrm{\mu }{x}^{a}a{y}^b}{x}\,1+\frac{\mathrm{\mu }{x}^a{y}^{b}b}{y}\,x^a{y}^b-U\right\}$

$\mathrm{solve}\left(\mathrm{eqns}\,\left[x\,y\,\mathrm{\μ}\right]\right)$

$\left[\left[x={\ⅇ}^{-\frac{\ln \left(\frac{b}{a}\right)b-\ln \left(U\right)}{b+a}}\,y={\ⅇ}^{\frac{\ln \left(\frac{b}{a}\right)a+\ln \left(U\right)}{b+a}}\,\mathrm{\mu }=-\frac{{\ⅇ}^{\frac{\ln \left(\frac{b}{a}\right)a+\ln \left(U\right)}{b+a}}}{Ub}\right]\right]$

$\mathrm{eqns}:=\left\{A\left(x\right)B\left(y\right)-U\,1\+\frac{\mathrm{\μ}A\left(x\right)aB\left(y\right)}{x}\,1\+\frac{\mathrm{\μ}A\left(x\right)B\left(y\right)b}{y}\right\}$

$\mathrm{eqns}≔\left\{1+\frac{\mathrm{\mu }A\left(x\right)aB\left(y\right)}{x}\,1+\frac{\mathrm{\mu }A\left(x\right)B\left(y\right)b}{y}\,A\left(x\right)B\left(y\right)-U\right\}$

$\mathrm{solve}\left(\mathrm{eqns}\,\left[x\,y\,\mathrm{\μ}\right]\right)$