If you have symoblic toolbox,
You have your answer on this page : https://uk.mathworks.com/help/symbolic/solve-a-single-differential-equation.html#f1-11214
That would give something like that :
syms y(t)
Dy = diff(y);
ode = diff(y,t,2) + A*y^4 + B == 0
cond1 = Dy(0) == C*(y(0)-D); % Correction made after Torsten comment T(0) --> y(0)
cond2 = Dy(L) == E*(D-y(L)); % Correction made after Torsten comment T(L) --> y(L)
conds = [cond1 cond2];
ySol(x) = dsolve(ode,conds);
ySol = simplify(ySol)
