Solve the system of equations x^2=y^3,x^y=y^x in positive real numbers.

aplaya4lyfeSS1

aplaya4lyfeSS1

Answered question

2022-11-23

Solve the system of equations x 2 = y 3 , x y = y x in positive real numbers.

Answer & Explanation

greffhc

greffhc

Beginner2022-11-24Added 7 answers

ln x x = ln y y , 2 ln x = 3 ln y ,
then dividing memberwise,
2 x = 3 y ,
or
ln x + ln 2 = ln y + ln 3 .
You now have two linear equations in ln x and ln y.
ln x = 3 ln 3 2 , ln y = 2 ln 3 2 .
Moncelliqo4

Moncelliqo4

Beginner2022-11-25Added 1 answers

Write y = x 2 3 , then
x x 2 3 = x 2 x 3
so
x 2 3 = 2 x 3

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