Mathematica Asked on February 22, 2021
For real $x$ consider the trivial equation
$$|y'(x)|=-|x|.$$
Since the left side is always positive and the right always negative, there is no solution.
Let’s try
DSolve[Abs[y'[x]]==-Abs[x], y, x, Assumptions-> {x ∈ Reals}],
DSolve[Abs[y'[x]]==-RealAbs[x], y, x, Assumptions-> {x ∈ Reals}]
and
DSolve[Sqrt[y'[x]^2]==-Abs[x], y, x, Assumptions-> {x ∈ Reals}]
all giving the wrong result
{{y->Function[{x},Sign[x]/2 x^2+Subscript[[ConstantC], 1]]},{y->Function[{x},-Sign[x]/2 x^2+Subscript[[ConstantC], 1]]}}
At least
DSolve[RealAbs[y'[x]]==-RealAbs[x], y, x, Assumptions-> {x ∈ Reals}]
does return {}
.
Is this a bug or a feature?
Note that this is just one example. In any case when the equation is $f(y'(x))=…$ and $f$ contains square root or absolute value the results are wrong.
Edit: Originally, the equation $|y'(x)|=-e^x$ was used for the example, but as a user found out, in that particluar case there is a complex solution.
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