Short question about definition in ODEs Hello I just have a short question about a remark made in my first class of intro ODE. My Professor was just motivating with a simple example, he wrote, $$\frac{dy}{dx}=y(x)$$
So of course it was clear that he was referring in general to $y(x)=ce^{x}$ where $c \in \mathbb{R}$
and then he also added and for all $ \ x \in \mathbb{R}$.
Now here is where I got briefly confused , because I thought for example consider $y(x)=e^9$, then $\frac{dy}{dx}=0 \neq e^{9}$
So what I took this to mean is that $y(x)=ce^x$ is the solution, and if you evaluate it at any x the equation is true, i.e. simply because $e^x$ was the solution? I just want to make sure that is what is meant in this context, or if there was some error in my understanding/notation?
Thanks all.
 A: $e^9$ is a constant, and so is not of the form $ce^x$.
It is true that $y = c e^x$ is a solution to the differential equation, where $c \in \mathbb R$ is any constant.
But this is a function of $x$ and so cannot give a derivative of zero unless it is a constant function.
It is analogous to saying:
Let $z(t) = t^2$ for $t \in \mathbb R$. Then $z(t) = 9^2$.
This is false; you can say $z(9) = 9^2$, for example, but not $z(t) = 9^2$.
This might help:
Consider $y_1 = e^x$. Then $y_1(9) = e^9$, and also $\left.{\dfrac {\mathrm dy_1}{\mathrm dx}}\right\vert_{x = 9} = e^9$.
This relationship between $y_1$ and $\dfrac {\mathrm dy_1}{\mathrm dx}$ would hold for all $x \in \mathbb R$.
A: There is a difference between $y(x)=ce^{x}$ and $y(x)=e^{9}$. The first one is an exponential function, but the second one is a horizontal line $y(x)=e^{9}=8103.08$. The derivative of the first one follow the rule the second one is just a constant number and will give a derivate of 0. 
When one says x is defined as a real number it does not mean that you can just replace it with a number like this  $y(x)=ce^{9}$. It means you can replace the input value, let's say $y(9)$ and solve for $y(x)$ at x=9.
A: The formulation 

"The solution is $y(x)=ce^x$ for some $c$ and all $x\in \Bbb R$" 

is short and casual for the more correct statement that 

"the solution of this ODE is a differentiable function $y:\Bbb R\to\Bbb R$ with function values $y(x)=ce^x$ (for all $x\in \Bbb R$)".

You will need to remember that a differential equation is a functional equation, i.e., its solutions are always functions. Sometimes, and often for textbook exercises, you can even give a computational algorithm for the values of this function.
