How to create a data table from an expression which take values from NDSolve?

Question

The bvp system of ode's I am having for this problem is,

ode1 = f'''[y] + a * f'[y]*f'[y] - f[y]*f''[y] == 0
ode2 = T''[y] + b *f[y]*T'[y] == 0

with boundary conditions

bcs = {f[0] == 0, f'[0] == 1, f'[L] == 0, T[0] == 1, T[L] == 0};

where L=5. Taking some random values for a, b and c,NDSolve report stiffness. The expression which should take values from NDSolve is

SE = f'[y]^2 + b*T[y]^2 + c*f[y]*T[y]; (*at y = 0*)

I want to vary a$\in[0.1,0.5]$, b$\in[0.1,1]$ and c$\in[0,2]$ by 0.1 simultaneously for both the system and the secondary expression SE to create a data table for SE of the form,

a b c SE

Can someone please help?

Edit

The above system is just a test case. So ignore the convergence warnings. I am only interested in generating the table.

Answer 1

A straightforward usage of ParametricNDSolveValue and Table:

L = 5;
bcs = {f[0] == 0, f'[0] == 1, f'[L] == 0, T[0] == 1, T[L] == 0};
ode1 = f'''[y] + a*f'[y]*f'[y] - f[y]*f''[y] == 0;
ode2 = T''[y] + b f[y] T'[y] == 0;

pa = ParametricNDSolveValue[{ode1, ode2, bcs}, {f, T}, {y, 0, L}, {a, b}];

midtable = Flatten[
   Table[{a -> aval, b -> bval, {f, T} -> pa[aval, bval] // Thread} // Flatten, {aval, 
     1/10, 5/10, 1/10}, {bval, 1/10, 1, 1/10}], 1];

SE = f'[y]^2 + b*T[y]^2 + c*f[y]*T[y] /. y -> 0;
rst = Table[{a, b, c, SE} /. midtable // Evaluate, {c, 0, 2, 1/10}]