# My question is about solving equation

Respected Sir, I am trying to get an equation of Qk in terms of Qp and k, for which
G[Q_] = 1/40 (x)^(1/5)*1/(12*Q^(1/5) (1 + Q)^(6/5)) (-32 (10125/(125 x)) Log[Q (1 + Q)^6] + 15 (1 + Q) (-4 + 20 Q -15 Q (1 + Q) Hypergeometric2F1[1, 8/5, 9/5, -Q])); Solve[G[Qk] - G[Qp] == y, Qk, Reals]

But the output is "Solve was unable to solve the system with inexact coefficients or the system obtained by direct rationalization of inexact numbers present in the system. Since many of the methods used by Solve require exact input, providing Solve with an exact version of the system may help." Please help me out. I have tried everything.

• (1) The message I get is "Solve::nsmet: This system cannot be solved with the methods available to Solve." (2) Mathematica has limited ability to solve transcendental equations with parameters. (I rather think mathematics at large has trouble with them.) (3) You can use FindRoot if you give numerical values to the parameters (x, y, Qp?). (4) You could also use InverseFunction, but I usually avoid it and use FindRoot. Jun 3, 2021 at 17:28
• Dear Michael E2, Thanks for your reply, values of x and Qp are 3.03*10^10 and 10^-1 respectively, but i have to get Qk as a function of y atleast. Any other way to solve the equation? Jun 4, 2021 at 1:42

I do not think that Solve or NSolve are able to handle this equation even with the values you provided in the comments.

Best thing I was able to produce was a single particular solution using FindInstance:

G[Q_] := 1/40 (x)^(1/5)*1/(12*Q^(1/5) (1 + Q)^(6/5)) (-32 (10125/(125 x)) Log[Q (1 + Q)^6] + 15 (1 + Q) (-4 + 20 Q -15 Q (1 + Q) Hypergeometric2F1[1, 8/5, 9/5, -Q]));
vars = {Qk,Qp,x,y}
vals = {Qp->(303/100)*10^10, x->10^-1}
eqn = FullSimplify[(G[Qk] - G[Qp] == y)/.vals]
inst = FindInstance[eqn, vars, Reals]
Print[inst]
Print[N[inst]]

gives

{{Qk -> 27, Qp -> -1/2, x -> -9/5, y -> (869780104257411228000947380*6^(2/5)*35^(4/5) - 224932050005938799999951*10^(1/5)*9180900000303^(4/5) + 5111580836587398075005567625000000*10^(1/5)*9180900000303^(4/5)*Hypergeometric2F1[1, 8/5, 9/5, -30300000000] - 18401691011714633070020043450*6^(2/5)*35^(4/5)*Hypergeometric2F1[1, 8/5, 9/5, -27] + 169344*10^(1/5)*9180900000303^(4/5)*Log[2] + 21168*10^(1/5)*9180900000303^(4/5)*Log[3] + 169344*10^(1/5)*9180900000303^(4/5)*Log[5] + 21168*10^(1/5)*9180900000303^(4/5)*Log[101] + 127008*10^(1/5)*9180900000303^(4/5)*Log[2789] + 127008*10^(1/5)*9180900000303^(4/5)*Log[10864109] - 100145257206610248000109080*6^(2/5)*35^(4/5)*Log[13011038208])/10904705784719782560011877600}}

or numerically:

{{Qk -> 27., Qp -> -0.5, x -> -1.8, y -> -7.60110441041372}}

Edit:

A little more insight can be gained by plotting y for particular choices of Qk.

If we define

nsol[n_] := NSolve[eqn&&Qk==n, vars, Reals]

it gives us the solution from FindInstance by calling

nsol[27]

{{Qk->27., y->-7.6011}}

We can now plot nsol:

Plot[(y/.nsol[n])//First,{n,0,30}]