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what is the terminology of this form of equation. It has only one variable, but with rational exponents, it can be positive, negative or fraction such as below:

$ax^2 +b/x^2 + c\sqrt{x} =0 $

I understand it is not a polynomial function. but what can I call it in proper math terminology? is it some kind of a one dimension ODE (Ordinary Differential Equations)?

is there any standard way to solve this type of function? how can I describe the complexity of this function? (like we describe polynomial with order). Many thanks!

Leo wang
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It's fair to call it an algebraic equation, since it consists of algebraic expressions. A standard approach toward solving it is to get rid of the square root symbol by setting $x=u^2$, which produces

$$au^4+{b\over u^4}+cu=0$$

and then clearing out the denominator to get a polynomial equation in $u$:

$$au^8+cu^5+b=0$$

There is no general formula for expressing the solutions $u$ in terms of $a$, $b$, and $c$, as there is for a quadratic equation; you can only solve it numerically (or sometimes exactly) for specific values of the coefficients.

Barry Cipra
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