Verify that the given function $y = Ax$ is a solution of the differential equation $xy' = y$ $(x \neq 0)$.
(A) Given function: $y = Ax$
Differentiating both sides with respect to $x$,we get:
$y' = \frac{d}{dx}(Ax) = A$
Now,substitute the values of $y$ and $y'$ into the given differential equation $xy' = y$:
$L.H.S. = xy' = x(A) = Ax$
$R.H.S. = y = Ax$
Since $L.H.S. = R.H.S.$,the given function $y = Ax$ is indeed a solution of the differential equation $xy' = y$.
Differentiating both sides with respect to $x$,we get:
$y' = \frac{d}{dx}(Ax) = A$
Now,substitute the values of $y$ and $y'$ into the given differential equation $xy' = y$:
$L.H.S. = xy' = x(A) = Ax$
$R.H.S. = y = Ax$
Since $L.H.S. = R.H.S.$,the given function $y = Ax$ is indeed a solution of the differential equation $xy' = y$.
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