Question:
$y=\mathrm{A} x \quad: \quad x y^{\prime}=y(x \neq 0)$
Solution:
$y=\mathrm{A} x$
Differentiating both sides with respect to x, we get:
$y^{\prime}=\frac{d}{d x}(\mathrm{Ax})$
$\Rightarrow y^{\prime}=\mathrm{A}$
Substituting the value of
in the given differential equation, we get:
L.H.S. $=x y^{\prime}=x \cdot \mathrm{A}=\mathrm{A} x=y=$ R.H.S.
Hence, the given function is the solution of the corresponding differential equation.