Xjyuk

I'm trying to have the a silimiar plot to the one Wolfram Alpha provides me here to the equation

$24.6347 left(246.347 x + sqrt(246.347 x + 20)^2 + 1 + 20right)^-a = 30.2879 left(-33.6532 x + sqrtx (1132.54 x - 20191.9) + 90001 + 300right)^-a$

for $a>0$. This equation however has no closed form solution for x in terms of a so I don't know how to achieve such a plot.

Minimal "Working" Example:

documentclass[11pt,a4paper]scrartcl
usepackagepgfplots
 pgfplotssetcompat=newest
 begindocument
 begintikzpicture


beginaxis[xlabel=a ,
 ylabel = x,
 xmin=0, xmax=10,
 ymin=0,ymax=1]
 addplot [
 domain=0:10, 
 samples=500, 
 color=blue,
 ]
 24.6347*(246.347*x+sqrt((246.347*x+20)^2+1)+20)^(-a)=30.2879*(-33.6532*x+ sqrt(x*(1132.54*x-20191.9)+90001)+300)^(-a); 
%switched the variables because gnuplot requires the variable x as input as far as i know
endaxis
endtikzpicture
enddocument

AFAIK you cannot really produce implicit plots but you can produce contour plots, which have the same information. So instead of plotting the solutions of A=B one plots the contours of A-B=0. You need to run with -shell-escape since this needs gnuplot (which your above example didn't need).

documentclass[11pt,a4paper]scrartcl
usepackagepgfplots
pgfplotssetcompat=newest
begindocument
begintikzpicture
beginaxis[view=090,ylabel=$a$,
 xlabel=$x$,xmin=0, xmax=1,ymin=0,ymax=1]
 addplot3 [contour gnuplot=levels=0,labels=false,
 domain=0:1,domain y=0:1] 
 24.6347*(246.347*x+sqrt((246.347*x+20)^2+1)+20)^(-y)
 -30.2879*(-33.6532*x+ sqrt(x*(1132.54*x-20191.9)+90001)+300)^(-y); 
endaxis
endtikzpicture
enddocument

In order to reproduce the Wolfram alpha plot you need to switch x and y back, and adjust the domains.

Actually, it does have a closed form solution, at least for a as a function of x.

run with xelatex:

documentclassarticle
usepackagepst-func
begindocument

pssetxunit=0.75
beginpspicture(-2,-1)(20,6)
psaxes[Dx=2]->(0,0)(-2,-0.5)(18,5)[$a$,0][$x$,90]
rput[lb](0,0)%
 pspicture*(0.1,0.1)(16,6)
 psplotImp[algebraic,linecolor=red,linewidth=1.5pt](0.05,0.05)(20,6)
 24.6347*(246.347*y + sqrt((246.347*y + 20)^2 + 1) + 20)^(-x)- 
 30.2879*(-33.6532*y + sqrt(y*(1132.54*y - 20191.9) + 90001) + 300)^(-x)
 endpspicture
endpspicture

enddocument