Question

. If xy + y = 1, find ay. dx

11th - 12th Class

Maths

Solution

101

4.0 (1 ratings)

( x^{y}+y^{x}=1 ) Let ( x^{y}=t ; y log x=log t ) ( Rightarrow frac{y}{x}+log x frac{d y}{d x}=frac{1}{t} frac{d t}{d x} ) ( Rightarrow frac{d t}{d x}=x^{y}left(frac{y}{x}+log x frac{d y}{d x}right) ) Let ( y^{x}=Delta j-x log y=log s ) ( Rightarrow ) leas.. ( log y+frac{x}{y} frac{d y}{d x}=frac{1}{5} frac{d s}{d x} ) ( Rightarrow frac{d s}{d x}=y^{x}left(log y+frac{x}{y} frac{d y}{d x}right) ) from ( (1), quad & x^{y}+y^{x}=1 Rightarrow t+s=1 ) ( Rightarrow frac{d t}{d x}+frac{d s}{d x}=0 ) ( Rightarrow y x^{y-1}+x^{y} log x d y+y^{x} log y+y^{x-1} x frac{d y}{d x}=0 ) ( Rightarrow frac{d y}{d x}=frac{-left(y^{x} log y+y x^{y-1}right)}{x^{y} log x+x y^{x-1}} )