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Consider this problem

    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]

It only appears the plot of u[y] but u'[y] is not shown. What's wrong?

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How about (I've set h=2 as an example)

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]

enter image description here

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