Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
$y = A x : x y^{'} = y (x \neq 0)$

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Solution

Let
$y = A x$
Differentiating both sides w.r.t. $x$ we get,
$y^{'} = (A x)^{'}$
$y^{'} = A$
Taking LHS
$x y^{'} = x A$
Putting $y = A x$
$x y^{'} = y$
Thus $L H S = R H S$
Hence verified.

Final answer:
Hence, the function $y = A x$ is a solution of the differential solution.

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