For simplicity sake,

.

What I cannot understand (similar to my other post) is why can you divide by

.

If you assume that

for any point

. Then you can solve it as a seperable

However, if you assume that there is a point where it is zero then you have two cases.

1)

is always zero for any point

. In that case,

are solutions.

2)

is zero and non-zero at some point. Then there exists an open interval where the function is non-zero. Thus, a solution as in the first example exists for this interval. In the other interval where it is zero is something else. Thus, the full solution is the piecewise defined function between these two, but that cannot be because that would imply non-differenciability, right?? Thus, that case is impossible.

Last comment I want to make is that,

is not a solution! You should say a function defined as

for

.