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

[sargondjani1]

I wonder if there is an analytical solution to:

b=a*x^-θ - 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.

 

[OmegaKV]

Analytic with respect to which variable?

 

[Ssnow]

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]

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.