Question

Solve:- x^y=y^x and x^2=y^3.​

Answer 1

x

y

=

y

x

admits as solution

x

=

y

Now applying

log

to the equations

{

x

y

=

y

x

x

2

=

y

3

-----(1)

we obtain

{

y

log

x

=

x

log

y

2

log

x

=

3

log

y

Dividing term to term we obtain the relationship

y

2

=

x

3

The solutions for (1) are obtained solving

{

x

=

y

x

2

=

y

3

and

{

x

2

=

y

3

y

2

=

x

3

so they are

⎛

⎜

⎜

⎝

x

=

0

y

=

0

x

=

1

y

=

1

x

=

27

8

y

=

9

4

⎞

⎟

⎟

⎠