2nd order non-linear differential equation

I have a heat transfer problem and I need to derive an 2nd order non-linear differential equation.

After replacing all the constants with A and B. My equation looks like this.

T'' + A(T)^4+ B= 0

Boundery conditions are T'(x=0)=C(T(0)-D), and T'(x=L)=E(D-T(L))

A,B,C,D,E are constants

And I need to solve for T(x).

Thank you in advance for your help.

Answer 1

[Florian Bidaud]

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)

[Torsten]

cond1 = Dy(0) == C*(y(0)-D);
cond2 = Dy(L) == E*(D-y(L));

ySol(t) = dsolve(ode,conds);

instead of

cond1 = Dy(0) == C*(T(0)-D);
cond2 = Dy(L) == E*(D-T(L));

ySol(x) = dsolve(ode,conds);

[Florian Bidaud]

@Torsten Indeed, thank you for spotting that