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

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.

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– user17762 May 14 '12 at 18:35
– user17762 May 14 '12 at 18:37

1
ok I deleted the other one. – Nikko May 14 '12 at 18:45

## 1 Answer

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 $T' = 0$.

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Thank you Robert. But can you explain me why you only take two combinations and not more? I mean why not $f(T,x)+A(T,x)−B(T,x)$ for instance ? – Nikko May 14 '12 at 18:47
$|t|$ is either $t$ or $-t$ (assuming we're talking about real numbers). – Robert Israel May 14 '12 at 19:07
Does not matter if A(T,x) and B(T,x) are negatives right?. Thank you very much! – Nikko May 15 '12 at 21:03
Right.  Why would it matter? – Robert Israel May 16 '12 at 0:27
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