Is $dX/dt=X(t)$ the correct ODE for $X(t)=e^t$?

Question

For a school project for chemistry I use systems of ODEs to calculate the concentrations of specific chemicals over time. Now I am wondering if

$$ \frac{dX}{dt} =X(t) $$

the same is as

$$ X(t)=e^t . $$

As far as I know, this should be correct, because the derivative of $ e^t $ is the same as the current value. Can anyone confirm that this is correct (or not)?

I already searched for it on the internet but can't really find any articles about this. Thanks!

Answer 1

The differential equation

$$ \frac{d X}{dt}=X(t)$$

has the general solution

$$X(t)=Ce^t$$

where $C \in \mathbb R.$

Answer 2

It is true that $$X(t)=e^t$$ is a solution of the differential equation $X'(t)=X(t)$. But we know from the theory that the solution must be a family of functions, depending on an arbitrary constant.

The usual way to solve this separable equation is by writing

$$\frac{dX}X=dt$$

and by indefinite integration,

$$\log X=t+c$$ or $$X=e^{t+c}=Ce^t.$$

---

As for all functions $f$, $(Cf(t))'=Cf'(t)$ (the differentiation operator is linear), you could have inferred that all $Ce^t$ are solutions of $X'(t)=X(t)$. But this does not guarantee yet that it is the most general solution.