a = 0.1;
    mu[y_]:=(1 + a*(6 (y/h)^2 - 1/2));
    m = -0.4; cost = -14.9; b=0.81;
    equ=D[(mu[y] (1 + b (u'[y])^2)^m *u'[y]), y]
    sol= NDSolve[{equ == cost, u[-1/2] == 0, u[1/2] == 0},
    u[y], {y, -1/2, 1/2}]
    Plot[Evaluate[{u[y], u'[y]} /.sol], {y, -0.5, 0.5},
    PlotRange -> All]

a = 0.1;
h = 2.;
mu[y_] := (1 + a*(6 (y/h)^2 - 1/2));
m = -0.4; cost = -14.9; b = 0.81;
equ = D[(mu[y] (1 + b (u'[y])^2)^m*u'[y]), y]
sol = NDSolve[{equ == cost, u[-1/2] == 0, u[1/2] == 0},
  u, {y, -1/2, 1/2}]
u = u /. sol[[1]]

    Plot[Evaluate[{u[y], u'[y]} /.sol], {y, -0.5, 0.5},
    PlotRange -> All]

sol = NDSolveValue[{equ == cost, u[-1/2] == 0, u[1/2] == 0}, u, {y, -1/2,1/2}]
Plot[{sol[y], sol'[y]}, {y, -0.5, 0.5}, PlotRange -> All]