Is $dX/dt=X(t)$ the correct ODE for $X(t)=e^t$?
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!
$\endgroup$2 Answers
$\begingroup$The differential equation
$$ \frac{d X}{dt}=X(t)$$
has the general solution
$$X(t)=Ce^t$$
where $C \in \mathbb R.$
$\endgroup$ 2 $\begingroup$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.
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