Question

For the following question verify that the given function (explicit or implicit) is a solution of the corresponding differential equation.

y=Ax and xy'=y $(x\ne 0)$

Answer 1

Given, y=Ax
On differentiating both sides w.r.r. x, we get
$y^{\prime }=\frac{d}{dx}\left(Ax\right)\Rightarrow y^{\prime }=A$
But we have to verify, $x{y}^{\prime }=y(x\ne 0)...\left(i\right)$
On substituting the value of y' in the Eq. (i), we get
$LHS=x{y}^{\prime }=x.A=A.x=y=RHS$
Hence, y=Ax is a solution of the given differential equation.