How to manage the absolute value on a differential equation $|T(x)'+A(T,x)+B(T,x)| = f(T,x)$

Question

Hi everyone I need to solve an equation of this type:

$|T(x)'+A(T,x)+B(T,x)| = f(T,x)$

with boundaries conditions.

The absolute value is my problem. Of course without it, the solution of these is well treated in the literature.

After search in the questions I found this: Differential equation with absolute value

So, can I do the same procedure? Or there is another way to solve this? Thanks.

Answer 1

Your equation does not determine $T'$ in terms of $T$ and $x$, so uniqueness of solutions may be a problem. You can say $T'$ is either $f(T,x) - A(T,x) - B(T,x)$ or $-f(T,x) - A(T,x) - B(T,x)$. Presumably there will be one region where it's the first and one region where it's the second, and (assuming $f,A,B$ are continuous) if you want $T'$ to exist everywhere it'll be impossible to switch between one and the other except when $T' = 0$.