no of solutions of the initial value problem? $x \dfrac{dy}{dx} = y , y (0) = 0, x \geq 0  .$
My Approach :
$\dfrac{dy}{y} = \dfrac{dx}{x},$ by variable separable method, we get
$lny = ln x +c $ and then raising e to both sides will get $ y=x+c.$  Then substituting the intial condtions, we get $c=-1$
so $y = x -1 $is the required solution.
But it's given that it has uncountable number of solution. I don't understand it.

If you solve the differential equation correctly, then
$$\ln |y|= \ln x + c = \ln e^c x,$$
and so
$$y=\pm e^c x,$$
not $y=x+c$. Then $e^c \cdot 0 = 0$ for all constant $C$, given initial value doesn't give us a useful information.
