Here is the code:
a = 4;
b = 10;
c = 3/10;
su = NDSolveValue[{b - c (1 - y/50) y u[y]^(1/4) - a u[y] +
1/8 y u'[y] + (y^2 u'[y]^2)/(16 u[y]) + (
y^2 (-3 u'[y]^2 + 4 u[y] u''[y]))/(32 u[y]) == 0,
u[0.6] == 200^(4/3), u'[50] == 0}, u, {y, 0.6, 50}, WorkingPrecision -> 50];
LogLogPlot[su[y], {y, 1, 50}, PlotRange -> All, AxesLabel -> {u, y}]
When I set c=3/10, there is no problem. However, I want to increase the value of c, say from 3/10 to 3 and 30. Then the following error encounters:
Power::infy: "Infinite expression 1/0 encountered"
Is this a problem of the equation itself? or is there any way for mathematica to find the solution when I increase the value of c to 3 and 30? Thanks!