Solution of certain forms of equations - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T12:27:43Z http://mathoverflow.net/feeds/question/113841 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/113841/solution-of-certain-forms-of-equations Solution of certain forms of equations RH 2012-11-19T16:15:03Z 2012-11-19T19:07:20Z <p>I ask about a possible method to find the solution of algebraic equations of the form</p> <p>$axⁿ+byⁿ+c=0$</p> <p>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.</p> http://mathoverflow.net/questions/113841/solution-of-certain-forms-of-equations/113845#113845 Answer by Robert Israel for Solution of certain forms of equations Robert Israel 2012-11-19T17:16:29Z 2012-11-19T19:07:20Z <p>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 $y = x^r$, $$n = \frac{\ln(-c/a)}{\ln(x)} + \sum_{k=0}^\infty \frac{(-c/a)^{kr}(b/c)^k}{k! \ln(x)} \prod_{j=1}^{k-1} (kr - j)$$</p>