# Analytical solution to b=a*x^-theta - x?

I wonder if there is an analytical solution to:

b=a*x - x

with a>0, b>0, x>0, θ>1

Erland
Do you by "analytical solution" mean a function built up from elementary functions? If so, the answer is almost certainly no, but I can't prove it.

Analytic with respect to which variable?

Ssnow
Gold Member
Do you want to search for ##x=\sum_{i=0}^{\infty}a_{i}y^{i}##? In this case put ##x## in ##bx^{\theta}=a-x^{\theta+1}## and try to solve a system (infinite) involving equations with the coefficients ##a_{i}##. If you want simply the solutions of ##bx^{\theta}-a+x^{\theta+1}=0##, you must start to consider different value of ##\theta##, remember that for equation of degree up or equal to ##5## there is not a solution given by radicals ...

mfb
Mentor
remember that for equation of degree up or equal to 55 there is not a solution given by radicals ...
That just gives two options for θ with an analytic solution.

Ssnow