Eta =  Phi[x] - a - b
linearisedBV = Collect[Normal[Series[(Exp[((1 - alpha) F Eta)/(R T)] -  Exp[ (-alpha F Eta)/(R T)]), {Phi[x], phistar, 1}]], Phi[x]]
eqn = Phi''[x] == k*j0*linearisedBV
soln = DSolve[{eqn, Phi'[0] == 0, Phi'[1] == 1}, Phi[x], x]

Eta =  Phi[x] - a - b;
eqn = Phi''[x] == k*j0*(Exp[((1 - alpha) F Eta)/(R T)] -  Exp[ (-alpha F Eta)/(R T)]);
soln = DSolve[{eqn, Phi'[0] == 0, Phi'[1] == 1}, Phi[x], x]

SetOptions[Solve, Method -> Reduce]

s = soln = DSolve[eqn, Phi[x], x]

(* Solve[
     Integrate[1/Sqrt[(2*E^((F*(a*(-1 + alpha) + (-1 + alpha)*b - alpha*K[1]))/(R*T))*
     ((-1 + alpha)*E^(((a + b)*F)/(R*T)) - alpha*E^((F*K[1])/(R*T)))*j0*k*R*T)/
     ((-1 + alpha)*alpha*F) + C[1]], {K[1], 1, Phi[x]}]^2 == (x + C[2])^2, Phi[x]] *)

ss = s[[1]] /. Power[z_, 2] -> z

(* Integrate[1/Sqrt[(2*E^((F*(a*(-1 + alpha) + (-1 + alpha)*b - alpha*K[1]))/(R*T))*
   ((-1 + alpha)*E^(((a + b)*F)/(R*T)) - alpha*E^((F*K[1])/(R*T)))*j0*k*R*T)/
   ((-1 + alpha)*alpha*F) + C[1]], {K[1], 1, Phi[x]}] == x + C[2] *)

sss = (Numerator[D[ss, x][[1]]])^2 == (Denominator[D[ss, x][[1]]])^2

(* Derivative[1][Phi][x]^2 ==
   (2*E^((F*(a*(-1 + alpha) + (-1 + alpha)*b - alpha*Phi[x]))/(R*T))*
   ((-1 + alpha)*E^(((a + b)*F)/(R*T)) - alpha*E^((F*Phi[x])/(R*T)))*j0*k*R*T)/
   ((-1 + alpha)*alpha*F) + C[1 *)

First@Solve[sss /. x -> 0 /. Phi'[0] -> 0, C[1]]

(* {C[1] -> (-2*E^((F*(a*(-1 + alpha) + (-1 + alpha)*b - alpha*Phi[0]))/(R*T))*
    (-E^(((a + b)*F)/(R*T)) + alpha*E^(((a + b)*F)/(R*T)) - alpha*E^((F*Phi[0])/(R*T)))*j0*
    k*R*T)/((-1 + alpha)*alpha*F)} *)

1 == (sss[[2]] /. x -> 1) - (sss[[2]] /. x -> 0)

(* 1 == (-2*E^((F*(a*(-1 + alpha) + (-1 + alpha)*b - alpha*Phi[0]))/(R*T))*
   ((-1 + alpha)*E^(((a + b)*F)/(R*T)) - alpha*E^((F*Phi[0])/(R*T)))*j0*k*R*T)/
   ((-1 + alpha)*alpha*F) +
   (2*E^((F*(a*(-1 + alpha) + (-1 + alpha)*b - alpha*Phi[1]))/(R*T))*
   ((-1 + alpha)*E^(((a + b)*F)/(R*T)) - alpha*E^((F*Phi[1])/(R*T)))*j0*k*R*T)/
   ((-1 + alpha)*alpha*F) *)

eqn = Phi''[x] == 2 k*j0*Sinh[Eta F/(2 R T)];
soln = Phi[x] /. DSolve[{eqn}, Phi[x], x] /.
    JacobiAmplitude[z1_, z2_] :> JacobiAmplitude[Simplify[z1], z2]

(* {(a F + b F + 4 I R T JacobiAmplitude[(I Sqrt[F] Sqrt[8 j0 k R T + F C[1]] (x + C[2]))/
       (4 R T), (16 j0 k R T)/(8 j0 k R T + F C[1])])/F,
    {(a F + b F - 4 I R T JacobiAmplitude[(I Sqrt[F] Sqrt[8 j0 k R T + F C[1]] (x + C[2]))/
       (4 R T), (16 j0 k R T)/(8 j0 k R T + F C[1])])/F} *)

Solve[{0 == D[soln[[1]], x] /. x -> 0, 1 == D[soln[[1]], x] /. x -> 1}, {C[1], C[2]}]

eqn = Phi''[x] == Exp[(1 - alpha) Phi[x]] - Exp[ -alpha Phi[x]];
ParametricNDSolveValue[{eqn, Phi'[0] == 0, Phi'[1] == 1}, Phi[0], x, {alpha}];
Plot[%[alpha], {alpha, 0, 1}, AxesLabel -> {alpha, "Phi[0]"}]