I am a student of physics, stuck with the general solution of an algebraic equation of the form: $$x^{n+2} + x^2 =f$$ I asked Mathematica, she couldn't solve it.
Is there any way to get a general solution for this type of equation (may be in terms some special function) in Mathematics literature?
Please forgive me, if the question is too naive!
If $n$ is small, you can use the usual polynomial tricks. If $n,x$ are large enough $x^{n+2}$ will be much larger than $x^2$. A good approximation to the solution will then be $x=f^{1/(n+2)}$. You can then do fixed point iteration using $x_{i+1}=(f-x_i^2)^{1/(n+2)}$ and iterate to convergence.