Just for a start, you can visualize the solution to the first equation in terms of M,f''[0]:
Manipulate[
sol = NDSolveValue[{
f'''[x] - (2 f'[x]^2 - 5/2 f[x] f''[x] + M*f''[x]) == 0,
f[0] == 0, f'[0] == 1, f''[0] == a},
f, {x, 0, xmax}];
Plot[sol[x], {x, 0, xmax}],
{{M, 1}, 0, 2},
{{a, -1.049, f''[0]}, -2, 2},
{xmax, 10, 100}]
