Can this dsolve code find solution recursively

syms x f f0(x) f1(x) f2(x) f3(x) f4(x) f5(x) f6(x)

eqn0 = diff(f1,3) == 0; cond0 = [f1(0) == 0, subs(diff(f1),0) == 0, subs(diff(f1),5) == 1 ];

f1 = dsolve(eqn0,cond0);

for m = 2:6

syms f_m

for k = 1:(m-1)

Df = diff(f); D2f = diff(Df); D3f = diff(D2f);

eqn(k) = D3f + (1/2)* f * D2f == 0;

cond(k) = [subs(f,0) == 0, subs(Df,0) == 0, subs(Df,5) == 0];

F = dsolve(eqn(k),cond(k)); f(k) = F.f(k)

end

end

disp([f2 f5 f])

%% I need (f1) to include in the loop to carry the solution and Please help me to run the code.

N.B: Two boundary conditions are different

PLEASE FOLLOW the attached pdf