Answer
False
Work Step by Step
$y=Ae^{x+b}$ must be solution to our desired ODE
So lets make ODE from given equation to see whether its true or not.
Differentiating both sides with respect to $x$
$\frac{dy}{dx} = \frac{d(Ae^{x+b})}{dx}$
Since $A$ is constant and differentiating $e^{x+b}$ using chain rule we get
$y'=Ae^{x+b}$
which is equal to $y$
hence $y'=y$ is the desired ODE
but it is of first degree, hence the answer to our question would be false.
