Solution of certain forms of equations

2012-11-19T16:15:03Z

I ask about a possible method to find the solution of algebraic equations of the form

$axⁿ+byⁿ+c=0$

where $a,b,c,x,y$ are real constants and $n$ is an integer. Maybe there is a simple method, but I cannot find it.

Answer by Robert Israel for Solution of certain forms of equations

2012-11-19T17:16:29Z

If $\log_x(y) = j/k$ is rational, this reduces to a polynomial in $x^{1/k} = y^{1/j}$. Otherwise you're unlikely to get a closed form. You might use numerical methods, or a series expansion: if $y = x^r$, $$ n = \frac{\ln(-c/a)}{\ln(x)} + \sum_{k=0}^\infty \frac{(-c/a)^{kr}(b/c)^k}{k! \ln(x)} \prod_{j=1}^{k-1} (kr - j)$$

Solution of certain forms of equations - MathOverflow

Solution of certain forms of equations

Answer by Robert Israel for Solution of certain forms of equations