Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 4*x^4-5*x^2+1=0
-
Equation 5*x^2+7*x+2=0
-
Equation x^3=4x
-
Equation (x-4)^2-25=0
- Express {x} in terms of y where:
- -2*x+4*y=3
- -14*x-13*y=11
- -1*x-5*y=-1
- -12*x-6*y=-3
- Plot:
-
x^y-y^x
-
Identical expressions
- x^y-y^x= zero
- x to the power of y minus y to the power of x equally 0
- x to the power of y minus y to the power of x equally zero
- xy-yx=0
- x^y-y^x=O
-
Similar expressions
- x^y+y^x=0
x^y-y^x=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
The graph
Rapid solution
[src]
/ /-log(y) \\ / /-log(y) \\
|y*W|--------|| |y*W|--------||
| \ y /| | \ y /|
x1 = - re|-------------| - I*im|-------------|
\ log(y) / \ log(y) /
$$x_{1} = - \operatorname{re}{\left(\frac{y W\left(- \frac{\log{\left(y \right)}}{y}\right)}{\log{\left(y \right)}}\right)} - i \operatorname{im}{\left(\frac{y W\left(- \frac{\log{\left(y \right)}}{y}\right)}{\log{\left(y \right)}}\right)}$$
x1 = -re(y*LambertW(-log(y)/y)/log(y)) - i*im(y*LambertW(-log(y)/y)/log(y))
Sum and product of roots
[src]
sum
/ /-log(y) \\ / /-log(y) \\
|y*W|--------|| |y*W|--------||
| \ y /| | \ y /|
- re|-------------| - I*im|-------------|
\ log(y) / \ log(y) /
$$- \operatorname{re}{\left(\frac{y W\left(- \frac{\log{\left(y \right)}}{y}\right)}{\log{\left(y \right)}}\right)} - i \operatorname{im}{\left(\frac{y W\left(- \frac{\log{\left(y \right)}}{y}\right)}{\log{\left(y \right)}}\right)}$$
=
/ /-log(y) \\ / /-log(y) \\
|y*W|--------|| |y*W|--------||
| \ y /| | \ y /|
- re|-------------| - I*im|-------------|
\ log(y) / \ log(y) /
$$- \operatorname{re}{\left(\frac{y W\left(- \frac{\log{\left(y \right)}}{y}\right)}{\log{\left(y \right)}}\right)} - i \operatorname{im}{\left(\frac{y W\left(- \frac{\log{\left(y \right)}}{y}\right)}{\log{\left(y \right)}}\right)}$$
product
/ /-log(y) \\ / /-log(y) \\
|y*W|--------|| |y*W|--------||
| \ y /| | \ y /|
- re|-------------| - I*im|-------------|
\ log(y) / \ log(y) /
$$- \operatorname{re}{\left(\frac{y W\left(- \frac{\log{\left(y \right)}}{y}\right)}{\log{\left(y \right)}}\right)} - i \operatorname{im}{\left(\frac{y W\left(- \frac{\log{\left(y \right)}}{y}\right)}{\log{\left(y \right)}}\right)}$$
=
/ /-log(y) \\ / /-log(y) \\
|y*W|--------|| |y*W|--------||
| \ y /| | \ y /|
- re|-------------| - I*im|-------------|
\ log(y) / \ log(y) /
$$- \operatorname{re}{\left(\frac{y W\left(- \frac{\log{\left(y \right)}}{y}\right)}{\log{\left(y \right)}}\right)} - i \operatorname{im}{\left(\frac{y W\left(- \frac{\log{\left(y \right)}}{y}\right)}{\log{\left(y \right)}}\right)}$$
-re(y*LambertW(-log(y)/y)/log(y)) - i*im(y*LambertW(-log(y)/y)/log(y))