While working out a new model for heat transport, I've come up with an equation of the form
$\frac{dF}{dT}\left(\frac{dT}{dx}\right)^2+F\frac{d^2T}{dx^2}=0,$
subject to $T(0)=1$ and $T(1)=0$. In the case where $F$ is independent of the temperature we recover the classical heat equation for steady state heeat conduction and the temperature profile $T(x)=1-x$. However, in general $F$ does not have to be independent of the temperature.
Does anybody have any information about equations of this form? I haven't been able to find anything related to this equation.
In the minimal case, the equation is $$ F'(T(x))T'(x)^2+F(T(x))T''(x)=0 $$ Divide by $T'$ and $F$ and integrate to get $$ \ln(F(T(x)))+\ln(T'(x))=c $$ or $$ F(T(x))T'(x)=C $$ which can be integrated again, denoting with $\Phi$ one of the anti-derivatives of $F$, $$ \Phi(T(x))=Cx+D $$ With the boundary conditions, $Φ(1)=D$ and $Φ(0)=C+D$, so that you have to solve the implicit equation $$ Φ(T(x))=xΦ(0)+(1-x)Φ(1) $$ for $T(x)$.