# 2nd order non-linear differential equation

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DJOULDE Aristide on 22 Aug 2023
Edited: Florian Bidaud on 22 Aug 2023
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).

Florian Bidaud on 22 Aug 2023
Edited: Florian Bidaud on 22 Aug 2023
If you have symoblic toolbox,
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 on 22 Aug 2023
cond1 = Dy(0) == C*(y(0)-D);
cond2 = Dy(L) == E*(D-y(L));
ySol(t) = dsolve(ode,conds);
cond1 = Dy(0) == C*(T(0)-D);
cond2 = Dy(L) == E*(D-T(L));
ySol(x) = dsolve(ode,conds);
Florian Bidaud on 22 Aug 2023
@Torsten Indeed, thank you for spotting that