# Solving of Equation which contains Hypergeometric Function 2F1

I am trying to solve this equation where I need the solution of K in term of v

Solve[1 -
K - (54 (20 -
K) v (2 (-10 +
K) (5 (1300 - 10 K + 3 K^2 - 100 (2 + K)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
3 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
1, (-10 + K)/(-20 + K)] + (-13000 + 400 K - 75 K^2 +
6 K^3 + 400 (-25 + 7 K) -
10 (200 - 140 K + 21 K^2)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
3 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
2, (-10 + K)/(-20 + K)]) (((-20 +
K)^3 Hypergeometric2F1[-1 - Sqrt[3], -1 + Sqrt[3],
1, -(10/(-20 + K))])/(-10 + K)^3)))/((12 (-20 + K) (-10 +
K) (5 (1300 - 10 K + 3 K^2 - 100 (2 + K)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
3 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
1, (-10 + K)/(-20 + K)] + (-13000 + 400 K - 75 K^2 +
6 K^3 + 400 (-25 + 7 K) -
10 (200 - 140 K + 21 K^2)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
3 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
2, (-10 + K)/(-20 + K)]) (-(((20 -
K)^3 Hypergeometric2F1[-1 - Sqrt[3], -1 + Sqrt[3],
1, -(K/(-20 + K))])/(8 (-10 + K)^3))) +
1/4 (-3 (-20 + K) (-10 Sqrt[3] - Sqrt[(-10 + K)^2] +
Sqrt[3] K) (-10 Sqrt[3] + Sqrt[(-10 + K)^2] +
Sqrt[3] K) (-K (2 K - 2 (10 + K)) Hypergeometric2F1[
2 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
2 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 1, (
2 (-10 + K))/(-20 + K)] -
2 (K^2 - 2 K (10 + K) +
4 (100 - 10 K + K^2)) Hypergeometric2F1[
2 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
2 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 2, (
2 (-10 + K))/(-20 + K)]) -
2 (-10 +
K)^2 (-K (3 K^2 - 2 K (50 + K) +
4 (300 - 10 K + K^2)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
3 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 1, (
2 (-10 + K))/(-20 + K)] - (10 (-28 + K) K^2 + 3 K^3 +
8 K (1050 - 70 K + K^2) +
16 (-6000 + 750 K - 40 K^2 + K^3)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2],
3 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 2, (
2 (-10 + K))/(-20 + K)])) )) == 0, K]

But, everytime I am getting a statement Solve::nsmet: This system cannot be solved with the methods available to Solve.

Please give me any suggestion. In have tried to find solution by using 'FindRoot'. But the problem with this is, If I give some value of 'v' then only it gives root.

• Side remark: ["Avoid single-capital-letter names for your variables"]()mathematica.stackexchange.com/a/18395/4999, especially ones used by the system, such as K. (Execute ? K) Aug 6, 2019 at 16:34

Massages from Solve says: transcendental equation can't be solved analytically,but we can plot solution of function k[v].

eq = 1 - k - (108 (20 - k) (-20 + k)^3 v Hypergeometric2F1[-1 - Sqrt[
3], -1 + Sqrt[3],
1, -(10/(-20 +
k))] (5 (1300 - 10 k + 3 k^2 - 100 (2 + k)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
1, (-10 + k)/(-20 + k)] + (-13000 + 400 k - 75 k^2 + 6 k^3 +
400 (-25 + 7 k) -
10 (200 - 140 k + 21 k^2)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
2, (-10 + k)/(-20 + k)]))/((-10 +
k)^2 (-(1/(2 (-10 + k)^2))
3 (20 - k)^3 (-20 + k) Hypergeometric2F1[-1 - Sqrt[3], -1 +
Sqrt[3],
1, -(k/(-20 +
k))] (5 (1300 - 10 k + 3 k^2 -
100 (2 + k)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
1, (-10 + k)/(-20 + k)] + (-13000 + 400 k - 75 k^2 +
6 k^3 + 400 (-25 + 7 k) -
10 (200 - 140 k + 21 k^2)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
2, (-10 + k)/(-20 + k)]) +
1/4 (-3 (-20 + k) (-10 Sqrt[3] - Sqrt[(-10 + k)^2] +
Sqrt[3] k) (-10 Sqrt[3] + Sqrt[(-10 + k)^2] +
Sqrt[3] k) (-k (2 k - 2 (10 + k)) Hypergeometric2F1[
2 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
2 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 1, (
2 (-10 + k))/(-20 + k)] -
2 (k^2 - 2 k (10 + k) +
4 (100 - 10 k + k^2)) Hypergeometric2F1[
2 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
2 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 2, (
2 (-10 + k))/(-20 + k)]) -
2 (-10 +
k)^2 (-k (3 k^2 - 2 k (50 + k) +
4 (300 - 10 k + k^2)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 1, (
2 (-10 + k))/(-20 + k)] - (10 (-28 + k) k^2 + 3 k^3 +
8 k (1050 - 70 k + k^2) +
16 (-6000 + 750 k - 40 k^2 + k^3)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 2, (
2 (-10 + k))/(-20 + k)]))));
ContourPlot[eq == 0, {v, -10, 1}, {k, 0, 11}, FrameLabel -> Automatic]

Addendum to the answer @MariuszIwaniuk. It is possible to construct a function for calculating $$k(v)$$ with a given WorkingPrecision

eq = 1 - k - (108 (20 - k) (-20 + k)^3 v Hypergeometric2F1[-1 -
Sqrt[3], -1 + Sqrt[3],
1, -(10/(-20 + k))] (5 (1300 - 10 k + 3 k^2 -
100 (2 + k)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
1, (-10 + k)/(-20 + k)] + (-13000 + 400 k - 75 k^2 +
6 k^3 + 400 (-25 + 7 k) -
10 (200 - 140 k + 21 k^2)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
2, (-10 + k)/(-20 + k)]))/((-10 +
k)^2 (-(1/(2 (-10 + k)^2)) 3 (20 - k)^3 (-20 +
k) Hypergeometric2F1[-1 - Sqrt[3], -1 + Sqrt[3],
1, -(k/(-20 + k))] (5 (1300 - 10 k + 3 k^2 -
100 (2 + k)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
1, (-10 + k)/(-20 + k)] + (-13000 + 400 k - 75 k^2 +
6 k^3 + 400 (-25 + 7 k) -
10 (200 - 140 k + 21 k^2)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
2, (-10 + k)/(-20 + k)]) +
1/4 (-3 (-20 + k) (-10 Sqrt[3] - Sqrt[(-10 + k)^2] +
Sqrt[3] k) (-10 Sqrt[3] + Sqrt[(-10 + k)^2] +
Sqrt[3] k) (-k (2 k - 2 (10 + k)) Hypergeometric2F1[
2 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
2 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
1, (2 (-10 + k))/(-20 + k)] -
2 (k^2 - 2 k (10 + k) +
4 (100 - 10 k + k^2)) Hypergeometric2F1[
2 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
2 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
2, (2 (-10 + k))/(-20 + k)]) -
2 (-10 +
k)^2 (-k (3 k^2 - 2 k (50 + k) +
4 (300 - 10 k + k^2)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
1, (2 (-10 + k))/(-20 + k)] - (10 (-28 + k) k^2 +
3 k^3 + 8 k (1050 - 70 k + k^2) +
16 (-6000 + 750 k - 40 k^2 + k^3)) Hypergeometric2F1[
3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2],
2, (2 (-10 + k))/(-20 + k)]))));

f[x_, p_] :=
Block[{v = x, $$MinPrecision = p,$$MaxPrecision = p},
Chop[k /. FindRoot[eq == 0, {k, 1}, WorkingPrecision -> p]]]

Then, for example, we have

f[-5, 20]

(*Out[]= 0.040304506890271919987*)

We can combine two solutions

lst = Table[{x, f[x, 30]}, {x, -10, 0, 1}];

fig1 = ListPlot[lst, PlotStyle -> Red];
fig2 = ContourPlot[eq == 0, {v, -10, 0}, {k, 0, 1},
FrameLabel -> Automatic, PlotPoints -> 50];
Show[fig2, fig1]